💡Alright, so I have this problem about an odd function. Let me try to understand what it's asking and figure out how to solve it step by step. First, the problem says that ( f(x) ) is an odd function defined on all real numbers. I remember that an odd function has the property that ( f(-x) = -f(x) ) for every ( x ) in its domain. That means if I know the value of the function at some positive ( x ), I can find the value at the corresponding negative ( x ) by just taking the negative of the positive value.Next, it tells me that when ( x > 0 ), ( f(x) = log_{2}x ). Okay, so for positive inputs, the function is just the logarithm base 2 of ( x ). That makes sense. I know that ( log_{2}x ) is defined for ( x > 0 ), which aligns with the given condition.The first part of the problem asks me to find the value of ( f(-4) + f(0) ). Let's break this down. Starting with ( f(-4) ): Since ( f(x) ) is an odd function, ( f(-4) = -f(4) ). I know that ( f(4) ) is ( log_{2}4 ) because 4 is positive. Calculating that, ( log_{2}4 ) is 2 because ( 2^2 = 4 ). So, ( f(-4) = -2 ).Now, what about ( f(0) )? Hmm, the function is defined on all real numbers, so 0 is included. For odd functions, ( f(-0) = -f(0) ). But ( -0 ) is just 0, so this simplifies to ( f(0) = -f(0) ). The only number that is equal to its own negative is 0. Therefore, ( f(0) = 0 ).Adding these together, ( f(-4) + f(0) = -2 + 0 = -2 ). Okay, that seems straightforward.The second part of the problem is a bit trickier. It says, if ( f(a) > f(-a) ), then find the range of the real number ( a ). Let's think about this.Again, since ( f(x) ) is odd, ( f(-a) = -f(a) ). So substituting that into the inequality, we get ( f(a) > -f(a) ). Let's write that out:( f(a) > -f(a) )If I add ( f(a) ) to both sides, I get:( 2f(a) > 0 )Dividing both sides by 2:( f(a) > 0 )So, the inequality simplifies to ( f(a) > 0 ). Now, I need to find all real numbers ( a ) such that ( f(a) ) is positive.Let's consider two cases: when ( a ) is positive and when ( a ) is negative.Case 1: ( a > 0 )In this case, ( f(a) = log_{2}a ). We need ( log_{2}a > 0 ). Remember that ( log_{2}a > 0 ) when ( a > 1 ) because ( log_{2}1 = 0 ) and the logarithm increases as ( a ) increases beyond 1. So, for positive ( a ), the condition ( f(a) > 0 ) holds when ( a > 1 ).Case 2: ( a < 0 )Here, since ( a ) is negative, we can use the property of the odd function again. ( f(a) = -f(-a) ). But ( -a ) is positive, so ( f(-a) = log_{2}(-a) ). Therefore, ( f(a) = -log_{2}(-a) ).We need ( f(a) > 0 ), which translates to:( -log_{2}(-a) > 0 )Multiplying both sides by -1 (and remembering to reverse the inequality sign):( log_{2}(-a) < 0 )So, ( log_{2}(-a) < 0 ) implies that ( -a < 1 ) because ( log_{2}x < 0 ) when ( 0 < x < 1 ). Since ( -a ) is positive (because ( a ) is negative), this inequality makes sense.So, ( -a < 1 ) implies ( a > -1 ). But we also know that ( a < 0 ) in this case. Therefore, combining these two inequalities, we get:( -1 < a < 0 )So, for negative ( a ), the condition ( f(a) > 0 ) holds when ( a ) is between -1 and 0.Putting both cases together, the values of ( a ) that satisfy ( f(a) > f(-a) ) are those where ( a > 1 ) or ( -1 < a < 0 ).Wait, let me double-check that. If ( a ) is between -1 and 0, then ( f(a) = -log_{2}(-a) ). Since ( -a ) is between 0 and 1, ( log_{2}(-a) ) is negative, so ( f(a) ) is positive. That makes sense. And for ( a > 1 ), ( f(a) ) is positive because ( log_{2}a ) is positive. So, yes, that seems correct.Is there any other case I need to consider? Well, ( a = 0 ) gives ( f(0) = 0 ), which doesn't satisfy ( f(a) > 0 ). And ( a = 1 ) gives ( f(1) = 0 ), which also doesn't satisfy the inequality. So, the ranges are open intervals, not including the endpoints.Therefore, the range of ( a ) is ( a > 1 ) or ( -1 < a < 0 ).Let me just recap:1. Calculated ( f(-4) ) using the odd function property: ( f(-4) = -f(4) = -log_{2}4 = -2 ).2. Found ( f(0) = 0 ) because ( f(-0) = -f(0) ) implies ( f(0) = 0 ).3. Summed them to get ( f(-4) + f(0) = -2 ).4. For the inequality ( f(a) > f(-a) ), used the odd function property to simplify it to ( f(a) > 0 ).5. Considered cases for ( a > 0 ) and ( a < 0 ), leading to the ranges ( a > 1 ) and ( -1 < a < 0 ).I think that covers everything. I don't see any mistakes in my reasoning, so I feel confident about the solution.**Final Answer**The value of ( f(-4) + f(0) ) is boxed{-2}, and the range of ( a ) is boxed{a > 1 text{ or } -1 < a < 0}.

💡First, identify the original number of fish in the aquarium, which is 10.Next, determine the current number of fish in the aquarium, which is 13.To find out how many fish were added, subtract the original number of fish from the current number of fish.So, 13 minus 10 equals 3.Therefore, 3 fish were added to the aquarium.

💡Okay, so I have this problem here: If the decimal representation of (2^n) starts with 7, what is the first digit in the decimal representation of (5^n)? The options are A) 1, B) 3, C) 5, D) 7, E) 9.Hmm, let's try to figure this out step by step. I remember that logarithms can be useful when dealing with leading digits of numbers. Maybe I can use logarithms to find some relationship between (2^n) and (5^n).First, if (2^n) starts with 7, that means (2^n) is somewhere between (7 times 10^k) and (8 times 10^k) for some integer (k). So, mathematically, I can write this as:[7 times 10^k leq 2^n < 8 times 10^k]Taking the logarithm base 10 of all parts of this inequality should help. Let's do that:[log_{10}(7 times 10^k) leq log_{10}(2^n) < log_{10}(8 times 10^k)]Simplifying each term:[log_{10}(7) + log_{10}(10^k) leq n log_{10}(2) < log_{10}(8) + log_{10}(10^k)]Since (log_{10}(10^k) = k), this becomes:[log_{10}(7) + k leq n log_{10}(2) < log_{10}(8) + k]I know that (log_{10}(8)) is the same as (3 log_{10}(2)) because (8 = 2^3). So, substituting that in:[log_{10}(7) + k leq n log_{10}(2) < 3 log_{10}(2) + k]This gives me a range for (n log_{10}(2)). Now, I need to relate this to (5^n). Let's take the logarithm of (5^n):[log_{10}(5^n) = n log_{10}(5)]I remember that (log_{10}(2) + log_{10}(5) = log_{10}(10) = 1), so (log_{10}(5) = 1 - log_{10}(2)). Therefore:[n log_{10}(5) = n (1 - log_{10}(2)) = n - n log_{10}(2)]From the earlier inequality, I have:[log_{10}(7) + k leq n log_{10}(2) < 3 log_{10}(2) + k]Subtracting this from (n) gives me:[n - (3 log_{10}(2) + k) < n log_{10}(5) leq n - (log_{10}(7) + k)]Simplifying:[n - 3 log_{10}(2) - k < n log_{10}(5) leq n - log_{10}(7) - k]But (n - k) is just the integer part of (n log_{10}(2)), which is an integer. Let's denote (m = n - k), so:[m - 3 log_{10}(2) < n log_{10}(5) leq m - log_{10}(7)]Since (m) is an integer, the fractional part of (n log_{10}(5)) is between ( -3 log_{10}(2) ) and ( -log_{10}(7) ). But fractional parts are between 0 and 1, so I need to adjust this.Wait, maybe I should think about the fractional part of (n log_{10}(2)). Let me denote the fractional part as ({n log_{10}(2)}). Then:[{n log_{10}(2)} = n log_{10}(2) - k]From the inequality:[log_{10}(7) leq {n log_{10}(2)} < 3 log_{10}(2)]Calculating the numerical values:[log_{10}(7) approx 0.8451][3 log_{10}(2) approx 3 times 0.3010 = 0.9030]So, the fractional part ({n log_{10}(2)}) is between approximately 0.8451 and 0.9030.Now, since (n log_{10}(5) = n - n log_{10}(2)), the fractional part of (n log_{10}(5)) is (1 - {n log_{10}(2)}) because subtracting a number from an integer doesn't change the fractional part, but here we're subtracting a non-integer. Wait, actually, let me think carefully.If (n log_{10}(5) = n - n log_{10}(2)), then the integer part is (n - lfloor n log_{10}(2) rfloor - 1) and the fractional part is (1 - {n log_{10}(2)}). Hmm, maybe not. Let's consider an example.Suppose (n log_{10}(2) = k + f), where (k) is an integer and (0 leq f < 1) is the fractional part. Then:[n log_{10}(5) = n - (k + f) = (n - k) - f]So, the integer part is (n - k - 1) and the fractional part is (1 - f). Therefore, the fractional part of (n log_{10}(5)) is (1 - f).Given that (f) is between 0.8451 and 0.9030, (1 - f) is between approximately 0.097 and 0.1549.So, the fractional part of (n log_{10}(5)) is between 0.097 and 0.1549. To find the first digit of (5^n), we look at (10^{text{fractional part}}).Calculating (10^{0.097}) and (10^{0.1549}):[10^{0.097} approx 1.25][10^{0.1549} approx 1.43]So, (5^n) starts with a number between approximately 1.25 and 1.43, which means the first digit is 1.Wait, but let me double-check. If the fractional part is between 0.097 and 0.1549, then (10^{text{fractional part}}) is between (10^{0.097}) and (10^{0.1549}). Calculating these:Using a calculator:[10^{0.097} approx 1.25][10^{0.1549} approx 1.43]Yes, so the leading digit is 1. Therefore, the first digit of (5^n) is 1.I think that makes sense. So, the answer should be A) 1.

💡Okay, so I have this problem where the first four terms of a geometric sequence are given as ( x + y ), ( x - y ), ( xy ), and ( frac{x}{y} ). I need to find the fifth term. Hmm, let me think about how to approach this.First, I remember that in a geometric sequence, each term is found by multiplying the previous term by a constant ratio, which I'll call ( r ). So, the ratio between the second and first term should be the same as the ratio between the third and second term, and so on.Let me write down the ratios:1. The ratio between the second and first term is ( frac{x - y}{x + y} ).2. The ratio between the third and second term is ( frac{xy}{x - y} ).3. The ratio between the fourth and third term is ( frac{frac{x}{y}}{xy} ).Since all these ratios are equal to ( r ), I can set them equal to each other.Starting with the first two ratios:[ frac{x - y}{x + y} = frac{xy}{x - y} ]Hmm, let me cross-multiply to solve for ( x ) and ( y ):[ (x - y)^2 = xy(x + y) ]Expanding both sides:[ x^2 - 2xy + y^2 = x^2y + xy^2 ]Let me bring all terms to one side:[ x^2 - 2xy + y^2 - x^2y - xy^2 = 0 ]Factor terms where possible:[ x^2(1 - y) - xy(2 + y) + y^2 = 0 ]This looks a bit complicated. Maybe I can factor differently or find a relationship between ( x ) and ( y ).Wait, let me try another approach. Let me denote ( x = ky ) where ( k ) is some constant. Maybe this substitution will simplify things.So, substituting ( x = ky ) into the equation:[ (ky - y)^2 = ky cdot y (ky + y) ]Simplify:[ (y(k - 1))^2 = ky^2 cdot y(k + 1) ][ y^2(k - 1)^2 = ky^3(k + 1) ]Divide both sides by ( y^2 ) (assuming ( y neq 0 )):[ (k - 1)^2 = ky(k + 1) ]Expand the left side:[ k^2 - 2k + 1 = k^2y + ky ]Hmm, this still seems complex. Maybe I should look at another ratio.Let me consider the ratio between the third and fourth terms:[ frac{frac{x}{y}}{xy} = frac{1}{y^2} ]Wait, that's not right. Let me recalculate:[ frac{frac{x}{y}}{xy} = frac{x}{y} cdot frac{1}{xy} = frac{1}{y^2} ]So, this ratio is ( frac{1}{y^2} ), which should also equal ( r ).So, now I have:1. ( r = frac{x - y}{x + y} )2. ( r = frac{xy}{x - y} )3. ( r = frac{1}{y^2} )Since all three expressions equal ( r ), I can set them equal to each other.First, set the first and third expressions equal:[ frac{x - y}{x + y} = frac{1}{y^2} ]Cross-multiplying:[ (x - y)y^2 = x + y ]Expanding:[ xy^2 - y^3 = x + y ]Bring all terms to one side:[ xy^2 - y^3 - x - y = 0 ]Factor terms:[ x(y^2 - 1) - y(y^2 + 1) = 0 ]Hmm, not sure if that helps. Maybe I can solve for ( x ) in terms of ( y ).From ( xy^2 - y^3 - x - y = 0 ):[ x(y^2 - 1) = y^3 + y ][ x = frac{y^3 + y}{y^2 - 1} ]Factor numerator and denominator:Numerator: ( y(y^2 + 1) )Denominator: ( (y - 1)(y + 1) )So,[ x = frac{y(y^2 + 1)}{(y - 1)(y + 1)} ]Okay, so I have ( x ) in terms of ( y ). Let me substitute this back into one of the earlier equations to find ( y ).Earlier, I had:[ (x - y)^2 = xy(x + y) ]Substituting ( x = frac{y(y^2 + 1)}{(y - 1)(y + 1)} ):First, compute ( x - y ):[ x - y = frac{y(y^2 + 1)}{(y - 1)(y + 1)} - y = frac{y(y^2 + 1) - y(y - 1)(y + 1)}{(y - 1)(y + 1)} ]Simplify the numerator:[ y(y^2 + 1) - y(y^2 - 1) = y^3 + y - y^3 + y = 2y ]So,[ x - y = frac{2y}{(y - 1)(y + 1)} ]Now, compute ( (x - y)^2 ):[ left( frac{2y}{(y - 1)(y + 1)} right)^2 = frac{4y^2}{(y - 1)^2(y + 1)^2} ]Next, compute ( xy(x + y) ):First, find ( x + y ):[ x + y = frac{y(y^2 + 1)}{(y - 1)(y + 1)} + y = frac{y(y^2 + 1) + y(y - 1)(y + 1)}{(y - 1)(y + 1)} ]Simplify the numerator:[ y(y^2 + 1) + y(y^2 - 1) = y^3 + y + y^3 - y = 2y^3 ]So,[ x + y = frac{2y^3}{(y - 1)(y + 1)} ]Now, compute ( xy ):[ xy = frac{y(y^2 + 1)}{(y - 1)(y + 1)} cdot y = frac{y^2(y^2 + 1)}{(y - 1)(y + 1)} ]So, ( xy(x + y) = frac{y^2(y^2 + 1)}{(y - 1)(y + 1)} cdot frac{2y^3}{(y - 1)(y + 1)} = frac{2y^5(y^2 + 1)}{(y - 1)^2(y + 1)^2} )Now, set ( (x - y)^2 = xy(x + y) ):[ frac{4y^2}{(y - 1)^2(y + 1)^2} = frac{2y^5(y^2 + 1)}{(y - 1)^2(y + 1)^2} ]Since the denominators are the same, set numerators equal:[ 4y^2 = 2y^5(y^2 + 1) ]Divide both sides by 2:[ 2y^2 = y^5(y^2 + 1) ]Bring all terms to one side:[ y^5(y^2 + 1) - 2y^2 = 0 ]Factor out ( y^2 ):[ y^2(y^5(y^2 + 1) - 2) = 0 ]So, either ( y^2 = 0 ) or ( y^5(y^2 + 1) - 2 = 0 )If ( y^2 = 0 ), then ( y = 0 ). But if ( y = 0 ), then the fourth term ( frac{x}{y} ) is undefined. So, ( y neq 0 ).Thus, solve:[ y^5(y^2 + 1) - 2 = 0 ]This seems complicated. Maybe I can find a rational root. Let me try ( y = 1 ):[ 1^5(1 + 1) - 2 = 2 - 2 = 0 ]Yes, ( y = 1 ) is a root. So, factor out ( (y - 1) ).Using polynomial division or synthetic division, divide ( y^7 + y^5 - 2 ) by ( y - 1 ).Wait, actually, the equation is ( y^5(y^2 + 1) - 2 = y^7 + y^5 - 2 = 0 ). So, we have ( y^7 + y^5 - 2 = 0 ).Using synthetic division for ( y = 1 ):Coefficients: 1 (y^7), 0 (y^6), 1 (y^5), 0 (y^4), 0 (y^3), 0 (y^2), 0 (y), -2 (constant)Bring down 1. Multiply by 1: 1. Add to next coefficient: 0 + 1 = 1. Multiply by 1: 1. Add to next coefficient: 1 + 1 = 2. Multiply by 1: 2. Add to next coefficient: 0 + 2 = 2. Multiply by 1: 2. Add to next coefficient: 0 + 2 = 2. Multiply by 1: 2. Add to next coefficient: 0 + 2 = 2. Multiply by 1: 2. Add to last term: -2 + 2 = 0. So, no remainder.Thus, ( y^7 + y^5 - 2 = (y - 1)(y^6 + y^5 + 2y^4 + 2y^3 + 2y^2 + 2y + 2) )So, the equation factors as:[ (y - 1)(y^6 + y^5 + 2y^4 + 2y^3 + 2y^2 + 2y + 2) = 0 ]We already found ( y = 1 ) is a solution. Now, check if ( y = 1 ) is valid.If ( y = 1 ), then from earlier ( x = frac{y(y^2 + 1)}{(y - 1)(y + 1)} ). But denominator becomes ( 0 ), so ( x ) is undefined. Thus, ( y = 1 ) is not a valid solution.Therefore, we need to solve ( y^6 + y^5 + 2y^4 + 2y^3 + 2y^2 + 2y + 2 = 0 ). Hmm, this is a sixth-degree polynomial, which might be difficult to solve. Maybe there are other rational roots.Using Rational Root Theorem, possible roots are ( pm1, pm2 ). Let me test ( y = -1 ):[ (-1)^6 + (-1)^5 + 2(-1)^4 + 2(-1)^3 + 2(-1)^2 + 2(-1) + 2 ][ = 1 - 1 + 2(1) + 2(-1) + 2(1) + (-2) + 2 ][ = 1 - 1 + 2 - 2 + 2 - 2 + 2 = 2 neq 0 ]Testing ( y = 2 ):This will be a large number, likely not zero.Testing ( y = -2 ):Similarly, large negative number, probably not zero.So, no rational roots. Maybe this polynomial has no real roots? Let me check the behavior.For large positive ( y ), the polynomial tends to positive infinity. For large negative ( y ), since the leading term is ( y^6 ), it also tends to positive infinity. At ( y = 0 ), the polynomial is 2. So, it's always positive? Let me check the derivative to see if it has any minima.But this might be too complicated. Maybe I made a mistake earlier in setting up the equations.Let me go back. Perhaps instead of substituting ( x = ky ), I should use the ratios directly.Given:1. ( r = frac{x - y}{x + y} )2. ( r = frac{xy}{x - y} )3. ( r = frac{frac{x}{y}}{xy} = frac{1}{y^2} )So, from 1 and 3:[ frac{x - y}{x + y} = frac{1}{y^2} ]Cross-multiplying:[ (x - y)y^2 = x + y ]Which simplifies to:[ xy^2 - y^3 = x + y ]Rearranged:[ xy^2 - x = y^3 + y ]Factor ( x ):[ x(y^2 - 1) = y(y^2 + 1) ]So,[ x = frac{y(y^2 + 1)}{y^2 - 1} ]Okay, so that's the same as before. Now, let's use the second ratio:[ r = frac{xy}{x - y} ]But we also have ( r = frac{1}{y^2} ). So,[ frac{xy}{x - y} = frac{1}{y^2} ]Cross-multiplying:[ xy cdot y^2 = x - y ][ xy^3 = x - y ]Bring all terms to one side:[ xy^3 - x + y = 0 ]Factor ( x ):[ x(y^3 - 1) + y = 0 ]But from earlier, ( x = frac{y(y^2 + 1)}{y^2 - 1} ). Substitute this in:[ frac{y(y^2 + 1)}{y^2 - 1}(y^3 - 1) + y = 0 ]Simplify:First, note that ( y^3 - 1 = (y - 1)(y^2 + y + 1) )So,[ frac{y(y^2 + 1)(y - 1)(y^2 + y + 1)}{y^2 - 1} + y = 0 ]Note that ( y^2 - 1 = (y - 1)(y + 1) ), so:[ frac{y(y^2 + 1)(y - 1)(y^2 + y + 1)}{(y - 1)(y + 1)} + y = 0 ]Cancel ( (y - 1) ):[ frac{y(y^2 + 1)(y^2 + y + 1)}{y + 1} + y = 0 ]Let me write this as:[ frac{y(y^2 + 1)(y^2 + y + 1)}{y + 1} = -y ]Multiply both sides by ( y + 1 ):[ y(y^2 + 1)(y^2 + y + 1) = -y(y + 1) ]Assuming ( y neq 0 ), divide both sides by ( y ):[ (y^2 + 1)(y^2 + y + 1) = -(y + 1) ]Expand the left side:[ (y^2 + 1)(y^2 + y + 1) = y^4 + y^3 + y^2 + y^2 + y + 1 = y^4 + y^3 + 2y^2 + y + 1 ]So,[ y^4 + y^3 + 2y^2 + y + 1 = -y - 1 ]Bring all terms to one side:[ y^4 + y^3 + 2y^2 + y + 1 + y + 1 = 0 ]Simplify:[ y^4 + y^3 + 2y^2 + 2y + 2 = 0 ]Hmm, this is a quartic equation. Let me check if it can be factored.Try rational roots: possible roots are ( pm1, pm2 ).Testing ( y = -1 ):[ (-1)^4 + (-1)^3 + 2(-1)^2 + 2(-1) + 2 = 1 - 1 + 2 - 2 + 2 = 2 neq 0 ]Testing ( y = -2 ):[ 16 - 8 + 8 - 4 + 2 = 14 neq 0 ]No rational roots. Maybe it factors into quadratics.Assume:[ (y^2 + ay + b)(y^2 + cy + d) = y^4 + y^3 + 2y^2 + 2y + 2 ]Multiply out:[ y^4 + (a + c)y^3 + (ac + b + d)y^2 + (ad + bc)y + bd ]Set coefficients equal:1. ( a + c = 1 )2. ( ac + b + d = 2 )3. ( ad + bc = 2 )4. ( bd = 2 )Looking for integer solutions. Possible pairs for ( b ) and ( d ) are (1,2), (2,1), (-1,-2), (-2,-1).Try ( b = 1 ), ( d = 2 ):Then, equation 3: ( a*2 + c*1 = 2 )From equation 1: ( c = 1 - a )Substitute into equation 3:[ 2a + (1 - a) = 2 ][ a + 1 = 2 ][ a = 1 ]Then, ( c = 0 )Check equation 2: ( (1)(0) + 1 + 2 = 3 neq 2 ). Not good.Try ( b = 2 ), ( d = 1 ):Equation 3: ( a*1 + c*2 = 2 )From equation 1: ( c = 1 - a )Substitute:[ a + 2(1 - a) = 2 ][ a + 2 - 2a = 2 ][ -a + 2 = 2 ][ -a = 0 ][ a = 0 ]Then, ( c = 1 )Check equation 2: ( 0*1 + 2 + 1 = 3 neq 2 ). Not good.Try ( b = -1 ), ( d = -2 ):Equation 3: ( a*(-2) + c*(-1) = 2 )From equation 1: ( c = 1 - a )Substitute:[ -2a - (1 - a) = 2 ][ -2a -1 + a = 2 ][ -a -1 = 2 ][ -a = 3 ][ a = -3 ]Then, ( c = 1 - (-3) = 4 )Check equation 2: ( (-3)(4) + (-1) + (-2) = -12 -1 -2 = -15 neq 2 ). Not good.Try ( b = -2 ), ( d = -1 ):Equation 3: ( a*(-1) + c*(-2) = 2 )From equation 1: ( c = 1 - a )Substitute:[ -a -2(1 - a) = 2 ][ -a -2 + 2a = 2 ][ a -2 = 2 ][ a = 4 ]Then, ( c = 1 - 4 = -3 )Check equation 2: ( 4*(-3) + (-2) + (-1) = -12 -2 -1 = -15 neq 2 ). Not good.So, no factorization with integer coefficients. Maybe this quartic has no real solutions? Let me check the discriminant or use calculus.Compute the derivative:[ f(y) = y^4 + y^3 + 2y^2 + 2y + 2 ][ f'(y) = 4y^3 + 3y^2 + 4y + 2 ]Set ( f'(y) = 0 ):[ 4y^3 + 3y^2 + 4y + 2 = 0 ]Again, difficult to solve. Maybe there's a minimum value.Compute ( f(y) ) at some points:At ( y = -2 ): ( 16 - 8 + 8 - 4 + 2 = 14 )At ( y = -1 ): ( 1 - 1 + 2 - 2 + 2 = 2 )At ( y = 0 ): ( 0 + 0 + 0 + 0 + 2 = 2 )At ( y = 1 ): ( 1 + 1 + 2 + 2 + 2 = 8 )So, the function is always positive, meaning ( f(y) = 0 ) has no real solutions. Therefore, the only real solution was ( y = 1 ), which was invalid because it made ( x ) undefined.Hmm, this suggests that there might be no real solutions for ( y ), which contradicts the problem statement that such a sequence exists. Maybe I made a mistake in my calculations.Let me go back to the beginning. The terms are ( x + y ), ( x - y ), ( xy ), ( frac{x}{y} ). Let me compute the ratios again.First ratio: ( r = frac{x - y}{x + y} )Second ratio: ( r = frac{xy}{x - y} )Third ratio: ( r = frac{frac{x}{y}}{xy} = frac{1}{y^2} )So, from first and third ratios:[ frac{x - y}{x + y} = frac{1}{y^2} ]Cross-multiplying:[ (x - y)y^2 = x + y ][ xy^2 - y^3 = x + y ]Rearranged:[ xy^2 - x = y^3 + y ][ x(y^2 - 1) = y(y^2 + 1) ][ x = frac{y(y^2 + 1)}{y^2 - 1} ]From second and third ratios:[ frac{xy}{x - y} = frac{1}{y^2} ]Cross-multiplying:[ xy cdot y^2 = x - y ][ xy^3 = x - y ]Rearranged:[ xy^3 - x = -y ][ x(y^3 - 1) = -y ][ x = frac{-y}{y^3 - 1} ]Now, I have two expressions for ( x ):1. ( x = frac{y(y^2 + 1)}{y^2 - 1} )2. ( x = frac{-y}{y^3 - 1} )Set them equal:[ frac{y(y^2 + 1)}{y^2 - 1} = frac{-y}{y^3 - 1} ]Assuming ( y neq 0 ), divide both sides by ( y ):[ frac{y^2 + 1}{y^2 - 1} = frac{-1}{y^3 - 1} ]Cross-multiplying:[ (y^2 + 1)(y^3 - 1) = - (y^2 - 1) ]Expand the left side:[ y^5 - y^2 + y^3 - 1 = -y^2 + 1 ]Bring all terms to one side:[ y^5 + y^3 - y^2 - 1 + y^2 - 1 = 0 ]Simplify:[ y^5 + y^3 - 2 = 0 ]So, ( y^5 + y^3 - 2 = 0 )Let me try to solve this quintic equation. Maybe it has a rational root.Possible rational roots: ( pm1, pm2 )Testing ( y = 1 ):[ 1 + 1 - 2 = 0 ]Yes, ( y = 1 ) is a root. So, factor out ( (y - 1) ).Using synthetic division on ( y^5 + y^3 - 2 ):Coefficients: 1 (y^5), 0 (y^4), 1 (y^3), 0 (y^2), 0 (y), -2Bring down 1. Multiply by 1: 1. Add to next coefficient: 0 + 1 = 1. Multiply by 1: 1. Add to next coefficient: 1 + 1 = 2. Multiply by 1: 2. Add to next coefficient: 0 + 2 = 2. Multiply by 1: 2. Add to next coefficient: 0 + 2 = 2. Multiply by 1: 2. Add to last term: -2 + 2 = 0.So, the polynomial factors as:[ (y - 1)(y^4 + y^3 + 2y^2 + 2y + 2) = 0 ]Again, ( y = 1 ) is a root, but as before, substituting ( y = 1 ) into ( x = frac{y(y^2 + 1)}{y^2 - 1} ) gives division by zero. So, ( y = 1 ) is invalid.Now, solve ( y^4 + y^3 + 2y^2 + 2y + 2 = 0 ). Let me check for real roots.Compute ( f(y) = y^4 + y^3 + 2y^2 + 2y + 2 )At ( y = -2 ): 16 - 8 + 8 - 4 + 2 = 14At ( y = -1 ): 1 - 1 + 2 - 2 + 2 = 2At ( y = 0 ): 0 + 0 + 0 + 0 + 2 = 2At ( y = 1 ): 1 + 1 + 2 + 2 + 2 = 8So, the function is always positive, meaning no real roots. Therefore, the only real solution is ( y = 1 ), which is invalid. This suggests that there might be no real solution, but the problem states that such a sequence exists. Maybe I made a mistake in my approach.Wait, perhaps I should consider that ( y ) could be negative. Let me try ( y = -1 ):From ( x = frac{y(y^2 + 1)}{y^2 - 1} ):[ x = frac{-1(1 + 1)}{1 - 1} ]Again, division by zero. Not good.Wait, maybe I should consider that ( y ) is a complex number, but the problem doesn't specify. It might be expecting a general expression in terms of ( y ).Wait, looking back at the original problem, it just asks for the fifth term, not necessarily to find ( x ) and ( y ). Maybe I can express the fifth term in terms of ( x ) and ( y ) without solving for them.Given that the fourth term is ( frac{x}{y} ), the fifth term would be ( frac{x}{y} cdot r ). From earlier, ( r = frac{1}{y^2} ). So, the fifth term is:[ frac{x}{y} cdot frac{1}{y^2} = frac{x}{y^3} ]But I need to express this in terms of ( x ) and ( y ) or find a relationship. Alternatively, maybe express ( x ) in terms of ( y ) from earlier.From ( x = frac{y(y^2 + 1)}{y^2 - 1} ), substitute into ( frac{x}{y^3} ):[ frac{frac{y(y^2 + 1)}{y^2 - 1}}{y^3} = frac{y(y^2 + 1)}{y^3(y^2 - 1)} = frac{y^2 + 1}{y^2(y^2 - 1)} ]Simplify:[ frac{y^2 + 1}{y^2(y^2 - 1)} = frac{y^2 + 1}{y^4 - y^2} ]Alternatively, factor numerator and denominator:Numerator: ( y^2 + 1 )Denominator: ( y^2(y - 1)(y + 1) )So, the fifth term is:[ frac{y^2 + 1}{y^2(y - 1)(y + 1)} ]But this seems complicated. Maybe there's a simpler way.Wait, from earlier, I had ( x(y^2 - 1) = y(y^2 + 1) ), so ( x = frac{y(y^2 + 1)}{y^2 - 1} ). Also, from ( xy^3 = x - y ), substituting ( x ):[ frac{y(y^2 + 1)}{y^2 - 1} cdot y^3 = frac{y(y^2 + 1)}{y^2 - 1} - y ]Simplify:[ frac{y^4(y^2 + 1)}{y^2 - 1} = frac{y(y^2 + 1) - y(y^2 - 1)}{y^2 - 1} ][ frac{y^4(y^2 + 1)}{y^2 - 1} = frac{y^3 + y - y^3 + y}{y^2 - 1} ][ frac{y^4(y^2 + 1)}{y^2 - 1} = frac{2y}{y^2 - 1} ]Multiply both sides by ( y^2 - 1 ):[ y^4(y^2 + 1) = 2y ][ y^6 + y^4 - 2y = 0 ]Factor:[ y(y^5 + y^3 - 2) = 0 ]So, ( y = 0 ) or ( y^5 + y^3 - 2 = 0 ). Again, ( y = 0 ) is invalid. So, ( y^5 + y^3 - 2 = 0 ). As before, ( y = 1 ) is a root, but it's invalid. The other roots are complex or irrational.This seems like a dead end. Maybe the problem expects an expression in terms of ( y ) without solving for it. From earlier, the fifth term is ( frac{x}{y^3} ), and ( x = frac{y(y^2 + 1)}{y^2 - 1} ), so:[ frac{x}{y^3} = frac{y(y^2 + 1)}{y^3(y^2 - 1)} = frac{y^2 + 1}{y^2(y^2 - 1)} ]Alternatively, factor numerator and denominator:[ frac{y^2 + 1}{y^2(y - 1)(y + 1)} ]But this doesn't simplify further. Maybe the answer is expected in terms of ( x ) and ( y ), but I'm not sure.Wait, looking back at the original problem, it just asks for the fifth term. Maybe I can express it in terms of the previous terms. The fifth term is the fourth term multiplied by ( r ). The fourth term is ( frac{x}{y} ), and ( r = frac{1}{y^2} ). So, fifth term is ( frac{x}{y} cdot frac{1}{y^2} = frac{x}{y^3} ).But without knowing ( x ) in terms of ( y ), I can't simplify further. Alternatively, from earlier, ( x = frac{y(y^2 + 1)}{y^2 - 1} ), so:[ frac{x}{y^3} = frac{y(y^2 + 1)}{y^3(y^2 - 1)} = frac{y^2 + 1}{y^2(y^2 - 1)} ]Alternatively, factor numerator and denominator:[ frac{y^2 + 1}{y^2(y - 1)(y + 1)} ]But this is as simplified as it gets. Maybe the answer is expected in this form.Alternatively, perhaps I made a mistake in assuming ( r = frac{1}{y^2} ). Let me double-check.From the third ratio:[ r = frac{frac{x}{y}}{xy} = frac{x}{y} cdot frac{1}{xy} = frac{1}{y^2} ]Yes, that's correct.So, the fifth term is ( frac{x}{y^3} ), which is ( frac{y^2 + 1}{y^2(y^2 - 1)} ).Alternatively, factor numerator and denominator:[ frac{y^2 + 1}{y^2(y - 1)(y + 1)} ]But this might not be necessary. Alternatively, express in terms of ( x ) and ( y ) as ( frac{x}{y^3} ).Wait, but the problem might expect a numerical answer, but since ( x ) and ( y ) are variables, it's likely expressed in terms of ( y ).Alternatively, maybe there's a simpler relationship I missed. Let me go back to the ratios.From the first ratio:[ r = frac{x - y}{x + y} ]From the third ratio:[ r = frac{1}{y^2} ]So,[ frac{x - y}{x + y} = frac{1}{y^2} ]Cross-multiplying:[ (x - y)y^2 = x + y ][ xy^2 - y^3 = x + y ]Rearranged:[ xy^2 - x = y^3 + y ][ x(y^2 - 1) = y(y^2 + 1) ][ x = frac{y(y^2 + 1)}{y^2 - 1} ]From the second ratio:[ r = frac{xy}{x - y} ]But ( r = frac{1}{y^2} ), so:[ frac{xy}{x - y} = frac{1}{y^2} ]Cross-multiplying:[ xy cdot y^2 = x - y ][ xy^3 = x - y ]Rearranged:[ xy^3 - x = -y ][ x(y^3 - 1) = -y ][ x = frac{-y}{y^3 - 1} ]Now, set the two expressions for ( x ) equal:[ frac{y(y^2 + 1)}{y^2 - 1} = frac{-y}{y^3 - 1} ]Assuming ( y neq 0 ), divide both sides by ( y ):[ frac{y^2 + 1}{y^2 - 1} = frac{-1}{y^3 - 1} ]Cross-multiplying:[ (y^2 + 1)(y^3 - 1) = - (y^2 - 1) ]Expand left side:[ y^5 - y^2 + y^3 - 1 = -y^2 + 1 ]Bring all terms to one side:[ y^5 + y^3 - 2 = 0 ]As before, ( y = 1 ) is a root, but invalid. The other roots are complex or irrational.This suggests that the only real solution is invalid, which contradicts the problem statement. Maybe the problem assumes ( y ) is a specific value, but it's not given. Alternatively, perhaps the fifth term can be expressed in terms of ( y ) without solving for it.Given that the fifth term is ( frac{x}{y^3} ), and ( x = frac{y(y^2 + 1)}{y^2 - 1} ), substitute:[ frac{x}{y^3} = frac{y(y^2 + 1)}{y^3(y^2 - 1)} = frac{y^2 + 1}{y^2(y^2 - 1)} ]Alternatively, factor:[ frac{y^2 + 1}{y^2(y - 1)(y + 1)} ]But this is as simplified as it gets. Alternatively, express in terms of ( x ) and ( y ) as ( frac{x}{y^3} ).Wait, but the problem might expect a numerical answer, but since ( x ) and ( y ) are variables, it's likely expressed in terms of ( y ).Alternatively, maybe there's a simpler relationship I missed. Let me think differently.From the first ratio:[ r = frac{x - y}{x + y} ]From the third ratio:[ r = frac{1}{y^2} ]So,[ frac{x - y}{x + y} = frac{1}{y^2} ]Cross-multiplying:[ (x - y)y^2 = x + y ][ xy^2 - y^3 = x + y ]Rearranged:[ xy^2 - x = y^3 + y ][ x(y^2 - 1) = y(y^2 + 1) ][ x = frac{y(y^2 + 1)}{y^2 - 1} ]Now, the fifth term is ( frac{x}{y} cdot r = frac{x}{y} cdot frac{1}{y^2} = frac{x}{y^3} )Substitute ( x ):[ frac{frac{y(y^2 + 1)}{y^2 - 1}}{y^3} = frac{y(y^2 + 1)}{y^3(y^2 - 1)} = frac{y^2 + 1}{y^2(y^2 - 1)} ]Simplify:[ frac{y^2 + 1}{y^2(y^2 - 1)} = frac{y^2 + 1}{y^4 - y^2} ]Alternatively, factor numerator and denominator:Numerator: ( y^2 + 1 )Denominator: ( y^2(y - 1)(y + 1) )So, the fifth term is:[ frac{y^2 + 1}{y^2(y - 1)(y + 1)} ]But this seems complicated. Maybe the answer is expected in terms of ( x ) and ( y ), but I'm not sure.Wait, perhaps I can express ( y^2 + 1 ) in terms of ( x ) and ( y ). From earlier, ( x = frac{y(y^2 + 1)}{y^2 - 1} ), so ( y^2 + 1 = frac{x(y^2 - 1)}{y} ). Substitute into the fifth term:[ frac{frac{x(y^2 - 1)}{y}}{y^2(y^2 - 1)} = frac{x(y^2 - 1)}{y cdot y^2(y^2 - 1)} = frac{x}{y^3} ]Which brings us back to the same expression. So, it seems the fifth term is ( frac{x}{y^3} ), but without knowing ( x ) in terms of ( y ), we can't simplify further.Alternatively, maybe the problem expects the answer in terms of ( y ) as ( frac{y^2 + 1}{y^2(y^2 - 1)} ).But looking back, the initial problem didn't specify whether ( x ) and ( y ) are real numbers or if they can be expressed in terms of each other. Given that, perhaps the fifth term is simply ( frac{x}{y^3} ), but I'm not sure.Wait, another approach: since it's a geometric sequence, the ratio ( r ) is consistent. So, the ratio between the fourth and third term is ( r = frac{frac{x}{y}}{xy} = frac{1}{y^2} ). Therefore, the fifth term is ( frac{x}{y} cdot frac{1}{y^2} = frac{x}{y^3} ).But without knowing ( x ) in terms of ( y ), I can't simplify further. Alternatively, from earlier, ( x = frac{y(y^2 + 1)}{y^2 - 1} ), so:[ frac{x}{y^3} = frac{y(y^2 + 1)}{y^3(y^2 - 1)} = frac{y^2 + 1}{y^2(y^2 - 1)} ]This is the simplest form I can get. So, the fifth term is ( frac{y^2 + 1}{y^2(y^2 - 1)} ).Alternatively, factor the denominator:[ frac{y^2 + 1}{y^2(y - 1)(y + 1)} ]But I think the first form is acceptable.So, after all this, the fifth term is ( frac{y^2 + 1}{y^2(y^2 - 1)} ).**Final Answer**The fifth term is boxed{dfrac{y^2 + 1}{y^2(y^2 - 1)}}.

💡Okay, so I have this geometry problem here. Let me read it again and try to understand what it's asking.Through a point P inside the triangle ABC, a line is drawn parallel to the base AB, dividing the triangle into two regions where the area of the smaller region (above the line) to the larger region (below the line) is in the ratio 1:3. If the altitude to AB has a length of 1, find the distance from P to AB.Alright, so we have triangle ABC with base AB. There's an altitude from C to AB, and that altitude is 1 unit long. Point P is somewhere inside the triangle, and we draw a line through P parallel to AB. This line divides the triangle into two regions: a smaller triangle on top and a quadrilateral below it. The areas of these two regions are in the ratio 1:3. We need to find the distance from point P to the base AB.Let me visualize this. Triangle ABC with base AB and height 1. Point P is somewhere inside, and we draw a line parallel to AB through P. This creates a smaller triangle on top, which is similar to triangle ABC because the line is parallel to AB. The area ratio between the smaller triangle and the quadrilateral below is 1:3. So, the smaller triangle has area 1 part, and the quadrilateral has area 3 parts. That means the total area of the triangle ABC is 4 parts.Since the line is parallel to AB, the smaller triangle is similar to triangle ABC. The ratio of their areas is 1:4 because the smaller triangle is 1 part and the whole triangle is 4 parts. I remember that when two similar figures have areas in the ratio of 1:4, the ratio of their corresponding linear measurements (like heights) is the square root of the area ratio. So, the square root of 1/4 is 1/2. That means the height of the smaller triangle is half the height of the entire triangle.Wait, the entire triangle has a height of 1, so the smaller triangle must have a height of 1/2. But hold on, the height of the smaller triangle is measured from the top vertex C down to the line we drew through P. So, if the height of the smaller triangle is 1/2, then the distance from P to AB would be the total height minus the height of the smaller triangle. That would be 1 - 1/2 = 1/2.But wait, that doesn't seem right because the area ratio is 1:3, not 1:4. Hmm, maybe I made a mistake here. Let me think again.The problem says the area of the smaller region to the larger region is 1:3. So, the smaller region is 1 part, and the larger region is 3 parts. That means the total area is 4 parts, as I thought before. So, the smaller triangle is 1/4 of the total area. Therefore, the area ratio is 1/4, which means the linear ratio is the square root of 1/4, which is 1/2. So, the height of the smaller triangle is 1/2 of the total height.But if the total height is 1, then the height of the smaller triangle is 1/2. So, the distance from P to AB is 1 - 1/2 = 1/2. But looking at the answer choices, 1/2 is option A, but I have a feeling that might not be correct because the area below the line is 3 parts, which is larger, so the distance from P to AB should be less than 1/2.Wait, maybe I got the ratio backwards. The area of the smaller region is 1, and the larger is 3, so the smaller triangle is 1/4 of the total area. Therefore, the height ratio is 1/2, meaning the height from the top is 1/2, so the distance from P to AB is 1 - 1/2 = 1/2. But that still gives me 1/2.Alternatively, maybe I should consider the ratio of the areas as the smaller region to the larger region is 1:3, so the area of the smaller triangle is 1/4 of the whole triangle. Therefore, the scaling factor for the sides is the square root of 1/4, which is 1/2. So, the height from the apex is 1/2, meaning the distance from P to AB is 1 - 1/2 = 1/2. Hmm, but I'm still getting 1/2, which is option A, but I think the answer is supposed to be 1/4.Wait, maybe I'm misunderstanding the problem. The line is drawn through P, which is inside the triangle, so the smaller region is above the line, and the larger region is below the line. So, the smaller triangle has area 1, and the quadrilateral has area 3. Therefore, the area of the smaller triangle is 1/4 of the total area. So, the height of the smaller triangle is sqrt(1/4) = 1/2. Therefore, the distance from P to AB is 1 - 1/2 = 1/2.But I'm confused because I think the answer should be 1/4. Maybe I need to approach this differently.Let me denote the distance from P to AB as h. Then, the height of the smaller triangle would be 1 - h. Since the triangles are similar, the ratio of their areas is (1 - h)^2 / 1^2 = (1 - h)^2. This ratio is equal to the area of the smaller triangle divided by the area of the whole triangle. Since the area ratio is 1:4, we have (1 - h)^2 = 1/4. Taking the square root of both sides, we get 1 - h = 1/2, so h = 1 - 1/2 = 1/2.Wait, that still gives me h = 1/2. But I think the answer is 1/4. Maybe I'm misapplying the ratio.Alternatively, perhaps the area ratio is 1:3, so the smaller area is 1 and the larger is 3, making the total area 4. Therefore, the area of the smaller triangle is 1/4 of the total area. So, the ratio of areas is 1/4, which means the ratio of heights is sqrt(1/4) = 1/2. Therefore, the height of the smaller triangle is 1/2, so the distance from P to AB is 1 - 1/2 = 1/2.But I'm still getting 1/2, which is option A. However, I think the correct answer is 1/4 because the line is closer to the base. Maybe I need to consider that the area below the line is 3/4, so the area above is 1/4. Therefore, the height ratio is sqrt(1/4) = 1/2, so the height from the apex is 1/2, meaning the distance from P to AB is 1 - 1/2 = 1/2.Wait, I'm going in circles here. Maybe I should set up the equation properly.Let me denote h as the distance from P to AB. Then, the height of the smaller triangle is 1 - h. The area of the smaller triangle is (1/2) * base * (1 - h). The base of the smaller triangle is proportional to (1 - h) because the triangles are similar. So, the base of the smaller triangle is (1 - h) times the base of the whole triangle.But the area of the whole triangle is (1/2) * base * 1. The area of the smaller triangle is (1/2) * (base * (1 - h)) * (1 - h) = (1/2) * base * (1 - h)^2.Therefore, the area ratio of the smaller triangle to the whole triangle is (1 - h)^2. We know this ratio is 1/4 because the smaller region is 1 part and the total is 4 parts. So, (1 - h)^2 = 1/4. Taking the square root, 1 - h = 1/2, so h = 1 - 1/2 = 1/2.Wait, so h is 1/2. That would mean the distance from P to AB is 1/2, which is option A. But I'm still unsure because I thought the answer was 1/4. Maybe I need to double-check.Alternatively, perhaps the area ratio is 1:3, so the smaller area is 1 and the larger is 3. Therefore, the area of the smaller triangle is 1/4 of the total area. So, the height ratio is sqrt(1/4) = 1/2. Therefore, the height from the apex is 1/2, so the distance from P to AB is 1 - 1/2 = 1/2.I think I'm consistent in getting 1/2, but I'm confused because I thought the answer was 1/4. Maybe I need to consider that the area below the line is 3/4, so the area above is 1/4. Therefore, the height from the apex is sqrt(1/4) = 1/2, so the distance from P to AB is 1 - 1/2 = 1/2.Wait, maybe I'm overcomplicating this. Let me try a different approach. Let's denote the distance from P to AB as h. The area of the quadrilateral below the line is 3/4 of the total area. The area of the quadrilateral can be found by subtracting the area of the smaller triangle from the total area.The area of the smaller triangle is (1/2) * base * (1 - h). The base of the smaller triangle is proportional to (1 - h), so it's (1 - h) times the base of the whole triangle. Let's denote the base of the whole triangle as b. Then, the area of the whole triangle is (1/2) * b * 1 = (1/2)b.The area of the smaller triangle is (1/2) * (b * (1 - h)) * (1 - h) = (1/2) * b * (1 - h)^2.The area of the quadrilateral is the total area minus the area of the smaller triangle: (1/2)b - (1/2)b(1 - h)^2 = (1/2)b[1 - (1 - h)^2].We know this area is 3/4 of the total area, so:(1/2)b[1 - (1 - h)^2] = (3/4)(1/2)bSimplify both sides:[1 - (1 - h)^2] = 3/4Expand (1 - h)^2:1 - (1 - 2h + h^2) = 3/4Simplify:1 - 1 + 2h - h^2 = 3/4Which becomes:2h - h^2 = 3/4Rearrange:h^2 - 2h + 3/4 = 0Multiply both sides by 4 to eliminate the fraction:4h^2 - 8h + 3 = 0Now, solve this quadratic equation for h:h = [8 ± sqrt(64 - 48)] / 8h = [8 ± sqrt(16)] / 8h = [8 ± 4] / 8So, h = (8 + 4)/8 = 12/8 = 3/2, which is greater than 1, so it's not possible.Or h = (8 - 4)/8 = 4/8 = 1/2.So, h = 1/2.Wait, so according to this, the distance from P to AB is 1/2, which is option A. But I thought earlier that the answer was 1/4. Maybe I was wrong earlier.But let me think again. If the area ratio is 1:3, then the smaller area is 1 and the larger is 3. So, the smaller triangle is 1/4 of the total area. Therefore, the height ratio is sqrt(1/4) = 1/2, so the height from the apex is 1/2, meaning the distance from P to AB is 1 - 1/2 = 1/2.Yes, that makes sense. So, the distance from P to AB is 1/2, which is option A.Wait, but I'm still confused because I thought the answer was 1/4. Maybe I need to check with an example.Suppose the total area is 4 units. The smaller triangle has area 1, and the quadrilateral has area 3. The height of the smaller triangle is 1/2, so the distance from P to AB is 1 - 1/2 = 1/2.Alternatively, if I consider the area below the line as 3, which is 3/4 of the total area, then the height from the base to the line would be sqrt(3/4) = sqrt(3)/2 ≈ 0.866, which is more than 1/2. But that contradicts the earlier result.Wait, no, that's not correct. The area ratio is 3/4, so the height ratio is sqrt(3/4) = sqrt(3)/2 ≈ 0.866, but that would be the height from the apex, not from the base. So, the distance from P to AB would be 1 - sqrt(3)/2 ≈ 1 - 0.866 ≈ 0.134, which is not one of the options.Hmm, this is confusing. Maybe I need to clarify.When we have similar triangles, the ratio of their areas is the square of the ratio of their corresponding heights. So, if the smaller triangle has area 1/4 of the total area, its height is 1/2 of the total height. Therefore, the distance from P to AB is 1 - 1/2 = 1/2.Alternatively, if the area below the line is 3/4, then the area above is 1/4, so the height from the apex is 1/2, so the distance from P to AB is 1 - 1/2 = 1/2.Wait, so both ways, I get 1/2. So, the answer should be 1/2, which is option A.But I'm still unsure because I thought the answer was 1/4. Maybe I need to consider that the area ratio is 1:3, so the smaller area is 1 and the larger is 3, making the total area 4. Therefore, the smaller triangle is 1/4 of the total area, so the height ratio is 1/2, so the distance from P to AB is 1 - 1/2 = 1/2.Yes, I think that's correct. So, the distance from P to AB is 1/2, which is option A.Wait, but in the initial problem, the altitude to AB is 1, so the total height is 1. If the smaller triangle has height 1/2, then the distance from P to AB is 1 - 1/2 = 1/2.Yes, that seems consistent. So, I think the answer is 1/2, which is option A.But wait, looking back at the problem, it says the area of the smaller region (above the line) to the larger region (below the line) is in the ratio 1:3. So, the smaller region is 1, and the larger is 3, making the total 4. Therefore, the smaller triangle is 1/4 of the total area, so the height ratio is 1/2, so the distance from P to AB is 1 - 1/2 = 1/2.Yes, that's correct. So, the answer is 1/2, which is option A.Wait, but I'm still confused because I thought the answer was 1/4. Maybe I need to think about it differently. Let me consider the area of the quadrilateral.The area of the quadrilateral is 3/4 of the total area. The area of the quadrilateral can be found by subtracting the area of the smaller triangle from the total area. So, if the area of the smaller triangle is 1/4, then the quadrilateral is 3/4.But the area of the quadrilateral can also be expressed as the area of the trapezoid formed by the line through P and the base AB. The area of a trapezoid is (1/2)*(sum of the two bases)*height. In this case, the two bases are AB and the line through P, and the height is the distance between them, which is h.Wait, but the line through P is parallel to AB, so the length of the line through P is proportional to the distance from P to AB. Let me denote the distance from P to AB as h. Then, the length of the line through P is (1 - h) times the length of AB because the triangles are similar.Wait, no, actually, the length of the line through P is proportional to the distance from the apex. So, if the height from the apex is (1 - h), then the length of the line through P is (1 - h) times the length of AB.But the area of the trapezoid (quadrilateral) is (1/2)*(AB + line through P)*h. Since AB is the base, and line through P is (1 - h)*AB, the area becomes (1/2)*(AB + (1 - h)AB)*h = (1/2)*AB*(1 + (1 - h))*h = (1/2)*AB*(2 - h)*h.But the area of the trapezoid is 3/4 of the total area, which is (3/4)*(1/2)*AB*1 = (3/8)*AB.So, setting up the equation:(1/2)*AB*(2 - h)*h = (3/8)*ABWe can cancel AB from both sides:(1/2)*(2 - h)*h = 3/8Multiply both sides by 2:(2 - h)*h = 3/4Expand:2h - h^2 = 3/4Rearrange:h^2 - 2h + 3/4 = 0Multiply by 4:4h^2 - 8h + 3 = 0Solve using quadratic formula:h = [8 ± sqrt(64 - 48)] / 8 = [8 ± sqrt(16)] / 8 = [8 ± 4]/8So, h = (8 + 4)/8 = 12/8 = 3/2 (invalid, since h cannot be more than 1) or h = (8 - 4)/8 = 4/8 = 1/2.So, h = 1/2.Therefore, the distance from P to AB is 1/2, which is option A.Wait, but earlier I thought the answer was 1/4. Maybe I was mistaken. According to this calculation, the distance is 1/2.But let me double-check. If h = 1/2, then the length of the line through P is (1 - 1/2) = 1/2 times AB. So, the trapezoid has bases AB and (1/2)AB, and height 1/2. The area would be (1/2)*(AB + (1/2)AB)*(1/2) = (1/2)*(3/2 AB)*(1/2) = (3/8)AB, which is indeed 3/4 of the total area (since total area is (1/2)AB*1 = (1/2)AB, and 3/8 AB is 3/4 of 1/2 AB).Yes, that checks out. So, the distance from P to AB is 1/2, which is option A.But wait, the answer choices are A. 1/2, B. 1/4, C. 3/4, D. 1/8, E. 1/3.So, according to my calculations, the answer is A. 1/2.But I'm still a bit confused because I thought the answer was 1/4. Maybe I need to think about it differently.Alternatively, perhaps the area ratio is 1:3, so the smaller area is 1 and the larger is 3, making the total area 4. Therefore, the smaller triangle is 1/4 of the total area, so the height ratio is sqrt(1/4) = 1/2, so the height from the apex is 1/2, meaning the distance from P to AB is 1 - 1/2 = 1/2.Yes, that's consistent. So, the answer is 1/2, which is option A.Wait, but I'm still unsure because I thought the answer was 1/4. Maybe I need to consider that the area below the line is 3/4, so the height from the base is sqrt(3/4) = sqrt(3)/2 ≈ 0.866, but that's not one of the options. Alternatively, maybe the height from the base is 1 - sqrt(1/4) = 1 - 1/2 = 1/2.Yes, that's correct. So, the distance from P to AB is 1/2.I think I've convinced myself that the answer is 1/2, which is option A.

💡Okay, so I've got this problem here about a line and a curve in the coordinate system. It's split into two parts. Let me try to figure out each part step by step.Starting with part (1): They give me a parametric equation for a line l. The parametric equations are x = 1 + t cos α and y = t sin α, where t is the parameter. I need to write the general equation of this line. Hmm, parametric equations usually express x and y in terms of a parameter, which here is t. To get the general equation, I think I need to eliminate the parameter t.So, from the first equation, x = 1 + t cos α, I can solve for t. Subtract 1 from both sides: x - 1 = t cos α. Then, divide both sides by cos α: t = (x - 1)/cos α. Now, plug this expression for t into the second equation, y = t sin α. So, y = [(x - 1)/cos α] * sin α. Simplify that: y = (x - 1) * (sin α / cos α). And sin α / cos α is tan α. So, y = tan α (x - 1). That should be the general equation of the line l. Alright, that seems straightforward. Now, moving on to the curve C. Its polar equation is given as ρ cos² θ - 4 sin θ = 0. I need to convert this into a Cartesian coordinate equation. I remember that in polar coordinates, ρ cos θ is x, and ρ sin θ is y. Also, ρ² = x² + y². Let me see how to manipulate the given equation. Starting with ρ cos² θ - 4 sin θ = 0. Maybe I can multiply both sides by ρ to make it easier to convert. So, ρ² cos² θ - 4 ρ sin θ = 0. Now, substitute ρ² cos² θ with x², because ρ cos θ = x, so ρ² cos² θ = x². Similarly, ρ sin θ is y. So, substituting, we get x² - 4y = 0. Therefore, the Cartesian equation is x² = 4y. That looks like a parabola opening upwards. Okay, part (1) seems done. Now, part (2) is a bit more involved. Let me read it again: Given point P(1,0). The polar coordinates of point M are (1, π/2). So, I need to find the Cartesian coordinates of M. In polar coordinates, (ρ, θ) = (1, π/2). Converting to Cartesian, x = ρ cos θ = 1 * cos(π/2) = 0. y = ρ sin θ = 1 * sin(π/2) = 1. So, point M is (0,1). Now, line l passes through M and intersects curve C at points A and B. The midpoint of AB is Q. I need to find |PQ|, the distance between P(1,0) and Q.So, first, since line l passes through M(0,1), and it's the same line l as given in part (1). Wait, but in part (1), the parametric equation was x = 1 + t cos α, y = t sin α. So, in that case, when t = 0, x = 1, y = 0, which is point P(1,0). So, line l passes through P and has direction determined by α. But now, it's passing through M(0,1). So, maybe the line l is the same, but with a different parameterization? Or perhaps α is different?Wait, the problem says "line l passes through M and intersects curve C at points A and B." So, line l is the same as in part (1), but now it's passing through M(0,1). So, perhaps we need to find the specific line l that passes through both P(1,0) and M(0,1). Wait, but in the parametric equation, when t = 0, it's at P(1,0). So, to pass through M(0,1), there must be some t value where x = 0 and y = 1. Let's see: x = 1 + t cos α = 0, so t cos α = -1. Similarly, y = t sin α = 1. So, t sin α = 1. So, from x: t = -1 / cos α. From y: t = 1 / sin α. Therefore, -1 / cos α = 1 / sin α. So, -sin α = cos α. Therefore, tan α = -1. So, α is -π/4 or 3π/4? Wait, but the inclination angle is usually between 0 and π, so it's 3π/4. But since tan α = -1, it's actually in the second quadrant, so α = 3π/4. But wait, inclination angle is measured from the positive x-axis, so 3π/4 is correct. So, the line l has an inclination angle α = 3π/4, which means tan α = -1. So, the general equation is y = tan α (x - 1) = -1*(x - 1) = -x + 1. So, the equation of line l is y = -x + 1. Wait, but in the parametric equation, we have x = 1 + t cos α, y = t sin α. Since α = 3π/4, cos α = -√2/2, sin α = √2/2. So, substituting, x = 1 + t*(-√2/2), y = t*(√2/2). So, the parametric equations are x = 1 - (√2/2) t, y = (√2/2) t. Now, this line intersects curve C, which is x² = 4y. So, let's substitute the parametric equations into x² = 4y. So, x = 1 - (√2/2) t, so x² = [1 - (√2/2) t]^2 = 1 - √2 t + ( (√2/2)^2 ) t² = 1 - √2 t + (2/4) t² = 1 - √2 t + (1/2) t². And 4y = 4*(√2/2) t = 2√2 t. So, setting x² = 4y: 1 - √2 t + (1/2) t² = 2√2 t. Bring all terms to one side: (1/2) t² - √2 t - 2√2 t + 1 = 0. Wait, that's not right. Let me do it step by step.x² = 1 - √2 t + (1/2) t²4y = 2√2 tSo, 1 - √2 t + (1/2) t² = 2√2 tSubtract 2√2 t from both sides:1 - √2 t - 2√2 t + (1/2) t² = 0Combine like terms:1 - 3√2 t + (1/2) t² = 0Multiply both sides by 2 to eliminate the fraction:2 - 6√2 t + t² = 0So, t² - 6√2 t + 2 = 0That's a quadratic in t. Let me write it as t² - 6√2 t + 2 = 0.Now, to find the points A and B, we need to solve for t. Let's use the quadratic formula.t = [6√2 ± sqrt( (6√2)^2 - 4*1*2 )]/2Calculate discriminant:(6√2)^2 = 36*2 = 724*1*2 = 8So, discriminant = 72 - 8 = 64So, sqrt(64) = 8Thus, t = [6√2 ± 8]/2 = [6√2 + 8]/2 or [6√2 - 8]/2Simplify:t = 3√2 + 4 or t = 3√2 - 4So, the two t values are t1 = 3√2 + 4 and t2 = 3√2 - 4.Now, the midpoint Q of AB would correspond to the average of t1 and t2. So, t_Q = (t1 + t2)/2 = [ (3√2 + 4) + (3√2 - 4) ] / 2 = (6√2)/2 = 3√2.So, the parameter t for point Q is 3√2.Now, let's find the coordinates of Q using the parametric equations.x = 1 - (√2/2)*t = 1 - (√2/2)*(3√2) = 1 - ( (√2 * 3√2 ) / 2 ) = 1 - ( (3*2) / 2 ) = 1 - 3 = -2Similarly, y = (√2/2)*t = (√2/2)*(3√2) = ( (√2 * 3√2 ) / 2 ) = (3*2)/2 = 3So, point Q is (-2, 3).Wait, but point P is (1,0). So, the distance |PQ| is the distance between (1,0) and (-2,3).Using the distance formula: sqrt[ (x2 - x1)^2 + (y2 - y1)^2 ] = sqrt[ (-2 - 1)^2 + (3 - 0)^2 ] = sqrt[ (-3)^2 + 3^2 ] = sqrt[9 + 9] = sqrt[18] = 3√2.So, |PQ| is 3√2.Wait, but let me double-check my steps to make sure I didn't make a mistake.First, converting the polar equation: ρ cos² θ - 4 sin θ = 0. I multiplied both sides by ρ to get ρ² cos² θ - 4 ρ sin θ = 0, which becomes x² - 4y = 0. That seems correct.Then, for part (2), converting M(1, π/2) to Cartesian: x = 1*cos(π/2)=0, y=1*sin(π/2)=1. So, M(0,1). Correct.Then, finding the inclination angle α. Since line l passes through P(1,0) and M(0,1), the slope is (1 - 0)/(0 - 1) = -1. So, tan α = -1, which gives α = 3π/4. Correct.Parametric equations: x = 1 + t cos(3π/4) = 1 - (√2/2)t, y = t sin(3π/4) = (√2/2)t. Correct.Substituting into x² = 4y: [1 - (√2/2)t]^2 = 4*(√2/2)t. Expanding, 1 - √2 t + (1/2)t² = 2√2 t. Bringing all terms to left: 1 - 3√2 t + (1/2)t² = 0. Multiply by 2: t² - 6√2 t + 2 = 0. Correct.Solving quadratic: t = [6√2 ± sqrt(72 - 8)]/2 = [6√2 ± 8]/2 = 3√2 ± 4. Correct.Midpoint t is (t1 + t2)/2 = (3√2 + 4 + 3√2 - 4)/2 = 6√2 /2 = 3√2. Correct.Coordinates of Q: x = 1 - (√2/2)(3√2) = 1 - 3 = -2. y = (√2/2)(3√2) = 3. So, Q(-2,3). Correct.Distance PQ: sqrt[(-2 -1)^2 + (3 -0)^2] = sqrt[9 +9] = sqrt[18] = 3√2. Correct.So, all steps seem correct. I think the answer is 3√2.