💡Okay, so I need to prove that if α and β satisfy the given inequalities and conditions, then α + β equals 3π/4. Let me start by understanding what's given.First, both α and β are between 0 and π/2. That means they are both acute angles. Also, tan α is 5, and cot β is 2/3. Hmm, I remember that cotangent is the reciprocal of tangent, so maybe I can find tan β from cot β.Let me write down what I know:- 0 ≤ α ≤ π/2- 0 ≤ β ≤ π/2- tan α = 5- cot β = 2/3Since cot β is 2/3, that means tan β is the reciprocal, so tan β = 3/2. Okay, that makes sense.Now, I need to find α + β. Maybe I can use the tangent addition formula. The formula is:tan(α + β) = (tan α + tan β) / (1 - tan α tan β)Let me plug in the values I have:tan(α + β) = (5 + 3/2) / (1 - 5 * 3/2)Let me compute the numerator and the denominator separately.Numerator: 5 + 3/2 = 5 + 1.5 = 6.5, which is 13/2.Denominator: 1 - (5 * 3/2) = 1 - 15/2 = (2/2 - 15/2) = -13/2.So, tan(α + β) = (13/2) / (-13/2) = -1.Hmm, tan(α + β) equals -1. But wait, α and β are both between 0 and π/2, so their sum α + β is between 0 and π. In this interval, the tangent of which angle is -1? I know that tan(3π/4) is -1 because 3π/4 is in the second quadrant where tangent is negative.So, α + β must be 3π/4.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I converted cot β to tan β correctly: cot β = 2/3 implies tan β = 3/2.2. I used the tangent addition formula correctly: (tan α + tan β)/(1 - tan α tan β).3. Calculated numerator: 5 + 1.5 = 6.5 = 13/2.4. Calculated denominator: 1 - 7.5 = -6.5 = -13/2.5. Divided them: (13/2)/(-13/2) = -1.6. Concluded that α + β = 3π/4 because tan(3π/4) = -1 and α + β is between 0 and π.Everything seems to check out. I don't see any errors in my reasoning. So, I think it's correct that α + β equals 3π/4.**Final Answer**boxed{alpha + beta = dfrac{3pi}{4}}

💡Alright, so I have this problem here: "A number raised to the power of 300 has 91 digits. What is the base of the number?" Hmm, okay, let's break this down step by step. I'm not entirely sure how to approach it, but I'll try to think it through.First, I know that when we talk about the number of digits in a number, we're essentially talking about its magnitude. For example, the number 100 has 3 digits, and 1000 has 4 digits. So, if a number has 91 digits, it's a pretty large number! Now, this number is the result of some base number raised to the power of 300. So, if I denote the base as "b", then the number in question is b^300.Now, I need to find out what "b" is. The problem is asking for the base of the number, which is "b". To find "b", I think I need to use logarithms because logarithms can help me figure out the exponent needed to reach a certain number. Specifically, I remember that the number of digits "d" in a number "N" can be found using the formula:d = floor(log10(N)) + 1Okay, so in this case, we have 91 digits, so:91 = floor(log10(b^300)) + 1Hmm, let's see. I can simplify this equation using logarithm properties. I recall that log(a^b) = b * log(a). So, applying that here:91 = floor(300 * log10(b)) + 1Now, I need to solve for "b". Let's isolate the floor function first:90 = floor(300 * log10(b))Since the floor function returns the greatest integer less than or equal to the number, I can approximate:90 ≈ 300 * log10(b)Now, dividing both sides by 300 to solve for log10(b):log10(b) ≈ 90 / 300log10(b) ≈ 0.3Alright, so log10(b) is approximately 0.3. To find "b", I need to find the antilogarithm, which means raising 10 to the power of 0.3:b ≈ 10^0.3I don't have a calculator handy, but I know that 10^0.3 is somewhere between 1 and 10. Specifically, I think 10^0.3 is approximately 2 because 10^0.3 is close to 2. Let me check that:10^0.3 ≈ 2Yes, that seems right. So, b is approximately 2. But wait, I should verify this because if I round down, the number of digits might be less than 91. Let me check if 2^300 actually has 91 digits.Calculating 2^300 directly is impractical because it's such a huge number, but I can use logarithms again to estimate the number of digits. Using the same formula:d = floor(log10(2^300)) + 1log10(2^300) = 300 * log10(2) ≈ 300 * 0.3010 ≈ 90.3So, d = floor(90.3) + 1 = 90 + 1 = 91Perfect! So, 2^300 does indeed have 91 digits. Therefore, the base of the number is 2.Wait a second, what if I tried a base slightly larger than 2, like 3? Let me see:log10(3^300) = 300 * log10(3) ≈ 300 * 0.4771 ≈ 143.13So, d = floor(143.13) + 1 = 143 + 1 = 144That's way more than 91 digits. So, 3 is too big. What about a base less than 2, like 1? Well, 1 raised to any power is always 1, which has only 1 digit. That's too small. So, 2 seems to be the smallest integer base that gives exactly 91 digits when raised to the 300th power.But just to be thorough, what if the base wasn't an integer? Let's say it's a decimal number. For example, if b = 1.5, then:log10(1.5^300) = 300 * log10(1.5) ≈ 300 * 0.1761 ≈ 52.83So, d = floor(52.83) + 1 = 52 + 1 = 53That's still way less than 91 digits. If I try b = 1.9:log10(1.9^300) = 300 * log10(1.9) ≈ 300 * 0.2788 ≈ 83.64d = floor(83.64) + 1 = 83 + 1 = 84Still not 91. What about b = 1.95:log10(1.95^300) ≈ 300 * log10(1.95) ≈ 300 * 0.2903 ≈ 87.09d = floor(87.09) + 1 = 87 + 1 = 88Closer, but still not 91. Let's try b = 1.99:log10(1.99^300) ≈ 300 * log10(1.99) ≈ 300 * 0.2989 ≈ 89.67d = floor(89.67) + 1 = 89 + 1 = 90Almost there. What about b = 1.995:log10(1.995^300) ≈ 300 * log10(1.995) ≈ 300 * 0.2998 ≈ 89.94d = floor(89.94) + 1 = 89 + 1 = 90Still 90 digits. To get to 91 digits, we need:floor(300 * log10(b)) + 1 = 91Which means:floor(300 * log10(b)) = 90So, 300 * log10(b) must be at least 90 but less than 91.Therefore:90 ≤ 300 * log10(b) < 91Dividing all parts by 300:0.3 ≤ log10(b) < 0.30333...Now, converting back from log10:10^0.3 ≤ b < 10^0.30333...Calculating 10^0.3 ≈ 2And 10^0.30333... ≈ 2.02So, b is between approximately 2 and 2.02.Since we're looking for an integer base, the only integer in this range is 2. Therefore, the base must be 2.I think that confirms it. The base is 2.

💡Okay, so I have this complex number z given by z = m² - m - 6 + (m² + 5m + 6)i, where m is a real number. There are two parts to this problem. First, I need to find the value of m such that z is purely imaginary. Hmm, okay, a complex number is purely imaginary if its real part is zero. So, that means the real part of z, which is m² - m - 6, must equal zero. Let me write that down: m² - m - 6 = 0. I need to solve this quadratic equation for m. To solve m² - m - 6 = 0, I can factor it. Let me see, I need two numbers that multiply to -6 and add up to -1. Hmm, 2 and -3? Yes, because 2 * (-3) = -6 and 2 + (-3) = -1. So, the equation factors to (m + 2)(m - 3) = 0. Therefore, m = -2 or m = 3. But wait, the problem says z is purely imaginary. That means not only the real part has to be zero, but the imaginary part must not be zero. So, I need to check if the imaginary part is non-zero for these values of m. The imaginary part is m² + 5m + 6. Let me plug in m = -2: (-2)² + 5*(-2) + 6 = 4 - 10 + 6 = 0. Oh, that's zero. So, m = -2 makes the imaginary part zero, which would make z a real number, not purely imaginary. So, m = -2 is not acceptable. Now, let's check m = 3: 3² + 5*3 + 6 = 9 + 15 + 6 = 30. That's definitely not zero. So, m = 3 is the valid solution. Therefore, the value of m is 3.Alright, moving on to the second part. The point corresponding to z is in the second quadrant of the complex plane. In the complex plane, the second quadrant corresponds to points where the real part is negative and the imaginary part is positive. So, I need both conditions: the real part of z is less than zero, and the imaginary part of z is greater than zero.Let me write down the inequalities:1. Real part: m² - m - 6 < 02. Imaginary part: m² + 5m + 6 > 0I need to solve these inequalities and find the range of m that satisfies both.Starting with the real part: m² - m - 6 < 0. I can factor this quadratic as (m + 2)(m - 3) < 0. To find where this product is negative, I can analyze the intervals determined by the roots m = -2 and m = 3.The critical points are at m = -2 and m = 3, which divide the real line into three intervals:1. m < -22. -2 < m < 33. m > 3I can test each interval:1. For m < -2, say m = -3: (-3 + 2)(-3 - 3) = (-1)(-6) = 6 > 0. So, positive in this interval.2. For -2 < m < 3, say m = 0: (0 + 2)(0 - 3) = 2*(-3) = -6 < 0. So, negative in this interval.3. For m > 3, say m = 4: (4 + 2)(4 - 3) = 6*1 = 6 > 0. So, positive in this interval.Since we want the inequality (m + 2)(m - 3) < 0, the solution is -2 < m < 3.Now, moving on to the imaginary part: m² + 5m + 6 > 0. Let's factor this quadratic as well. Looking for two numbers that multiply to 6 and add to 5, which are 2 and 3. So, it factors to (m + 2)(m + 3) > 0.Again, the critical points are m = -3 and m = -2. These divide the real line into three intervals:1. m < -32. -3 < m < -23. m > -2Testing each interval:1. For m < -3, say m = -4: (-4 + 2)(-4 + 3) = (-2)(-1) = 2 > 0. Positive.2. For -3 < m < -2, say m = -2.5: (-2.5 + 2)(-2.5 + 3) = (-0.5)(0.5) = -0.25 < 0. Negative.3. For m > -2, say m = 0: (0 + 2)(0 + 3) = 2*3 = 6 > 0. Positive.We want the inequality (m + 2)(m + 3) > 0, so the solution is m < -3 or m > -2.Now, combining both inequalities:From the real part, we have -2 < m < 3.From the imaginary part, we have m < -3 or m > -2.So, the overlap between these two solutions is where both conditions are satisfied. Let's see:- The real part requires -2 < m < 3.- The imaginary part requires m > -2 (since m < -3 doesn't overlap with -2 < m < 3).Therefore, the overlapping interval is -2 < m < 3.Wait, but hold on. Let me double-check. The imaginary part is positive when m < -3 or m > -2. But since the real part requires m to be between -2 and 3, the only overlapping region is m > -2. So, the combined solution is -2 < m < 3.Therefore, the range of m is -2 < m < 3.Let me just recap:1. For z to be purely imaginary, m must be 3.2. For z to be in the second quadrant, m must be between -2 and 3.I think that makes sense. I don't see any mistakes in my reasoning. The key was to factor the quadratics and then determine the intervals where the inequalities hold. Also, remembering that for a complex number to be purely imaginary, the real part must be zero and the imaginary part must be non-zero.**Final Answer**1. The value of ( m ) is (boxed{3}).2. The range of values for ( m ) is (boxed{-2 < m < 3}).

💡First, I recognize that both polygons have the same perimeter. Let's denote the side length of the second polygon as ( s ). Consequently, the side length of the first polygon, which is three times as long, would be ( 3s ).Next, I calculate the perimeter of the first polygon. Since it has 50 sides, each of length ( 3s ), the perimeter is:[50 times 3s = 150s]Since the perimeters are equal, the perimeter of the second polygon must also be ( 150s ). Let ( n ) represent the number of sides of the second polygon. The perimeter of the second polygon can be expressed as:[n times s]Setting the perimeters equal to each other gives:[n times s = 150s]Solving for ( n ) by dividing both sides by ( s ) yields:[n = 150]Therefore, the second polygon has 150 sides.

💡Okay, so I need to find the length of the tangent segment from the origin to the circle that passes through the points (4,3), (8,6), and (13,5). Hmm, let me think about how to approach this.First, I remember that the length of a tangent from a point to a circle can be found using the formula: if the circle has center (h, k) and radius r, and the point is (x1, y1), then the length of the tangent is sqrt[(x1 - h)^2 + (y1 - k)^2 - r^2]. In this case, the point is the origin (0,0), so the formula simplifies to sqrt[h^2 + k^2 - r^2].So, I need to find the center (h, k) and the radius r of the circle passing through the three given points. To find the circumcircle of a triangle, I can use the perpendicular bisectors of two sides to find the center.Let me label the points for clarity: let A be (4,3), B be (8,6), and C be (13,5). I'll start by finding the midpoints and slopes of two sides, say AB and AC.First, the midpoint of AB: the coordinates are ((4+8)/2, (3+6)/2) = (6, 4.5). The slope of AB is (6-3)/(8-4) = 3/4. Therefore, the slope of the perpendicular bisector of AB is the negative reciprocal, which is -4/3.So, the equation of the perpendicular bisector of AB is y - 4.5 = (-4/3)(x - 6). Let me write that out: y = (-4/3)x + (8) + 4.5. Wait, let me compute that again. Starting from point-slope form: y - 4.5 = (-4/3)(x - 6). So, y = (-4/3)x + (24/3) + 4.5. 24/3 is 8, so y = (-4/3)x + 8 + 4.5. 8 + 4.5 is 12.5, which is 25/2. So, the equation is y = (-4/3)x + 25/2.Now, let's find the perpendicular bisector of AC. The midpoint of AC is ((4+13)/2, (3+5)/2) = (17/2, 4). The slope of AC is (5-3)/(13-4) = 2/9. Therefore, the slope of the perpendicular bisector is -9/2.Using point-slope form, the equation of the perpendicular bisector of AC is y - 4 = (-9/2)(x - 17/2). Let me simplify that. First, distribute the slope: y - 4 = (-9/2)x + (9/2)*(17/2). Calculating the constant term: (9/2)*(17/2) = 153/4. So, y = (-9/2)x + 153/4 + 4. Converting 4 to fourths, that's 16/4, so y = (-9/2)x + (153 + 16)/4 = (-9/2)x + 169/4.Now, I have two equations of the perpendicular bisectors:1. y = (-4/3)x + 25/22. y = (-9/2)x + 169/4I need to solve these two equations to find the center (h, k) of the circle.Let me set them equal to each other:(-4/3)x + 25/2 = (-9/2)x + 169/4To eliminate fractions, let's multiply both sides by 12, which is the least common multiple of denominators 3, 2, and 4.12*(-4/3)x + 12*(25/2) = 12*(-9/2)x + 12*(169/4)Simplify each term:12*(-4/3)x = -16x12*(25/2) = 15012*(-9/2)x = -54x12*(169/4) = 507So, the equation becomes:-16x + 150 = -54x + 507Now, let's bring all terms to one side:-16x + 150 + 54x - 507 = 0( -16x + 54x ) + (150 - 507) = 038x - 357 = 0Solving for x:38x = 357x = 357 / 38Let me compute that. 357 divided by 38. 38*9=342, so 357-342=15. So, x = 9 + 15/38 = 9.3947 approximately.Now, plug x back into one of the equations to find y. Let's use the first equation: y = (-4/3)x + 25/2.So, y = (-4/3)*(357/38) + 25/2.First, compute (-4/3)*(357/38). Let's simplify 357/38. 38*9=342, so 357=38*9 +15, so 357/38=9 +15/38.So, (-4/3)*(9 +15/38) = (-4/3)*9 + (-4/3)*(15/38) = -12 - (60/114) = -12 - (10/19) = -12.5263 approximately.Now, 25/2 is 12.5. So, y = -12.5263 + 12.5 ≈ -0.0263. Hmm, that's very close to zero. Maybe I made a calculation error.Wait, let's do it more precisely.First, compute (-4/3)*(357/38):Multiply numerator: -4 * 357 = -1428Denominator: 3 * 38 = 114So, it's -1428 / 114. Let's simplify that.Divide numerator and denominator by 6: -1428 ÷6= -238, 114 ÷6=19. So, -238/19.238 divided by 19: 19*12=228, so 238-228=10. So, -238/19 = -12 -10/19.So, y = (-12 -10/19) + 25/2.Convert 25/2 to nineteenths: 25/2 = (25*9.5)/19 = 237.5/19. Wait, that's messy. Alternatively, let's find a common denominator.Convert both terms to have denominator 38.-12 -10/19 = -12 - (20/38) = -12.526325/2 = 475/38 ≈12.5So, y = (-12.5263) + 12.5 = -0.0263. Hmm, that's approximately zero. Maybe it's exactly -10/380 or something? Wait, let's do it fraction-wise.So, y = (-238/19) + (25/2). Let's find a common denominator, which is 38.-238/19 = (-238*2)/38 = -476/3825/2 = (25*19)/38 = 475/38So, y = (-476/38) + (475/38) = (-476 + 475)/38 = (-1)/38So, y = -1/38. That's approximately -0.0263, which matches the earlier decimal.So, the center of the circle is at (357/38, -1/38). Let me write that as fractions:x = 357/38, y = -1/38.Now, let's compute the radius r. The radius is the distance from the center to any of the points A, B, or C. Let's pick point A (4,3).Compute the distance squared between (357/38, -1/38) and (4,3):First, convert 4 and 3 to over 38:4 = 152/38, 3 = 114/38.So, the differences:x: 357/38 - 152/38 = (357 - 152)/38 = 205/38y: -1/38 - 114/38 = (-1 - 114)/38 = -115/38So, distance squared is (205/38)^2 + (-115/38)^2.Compute 205^2: 205*205. Let's compute 200^2 + 2*200*5 +5^2 = 40000 + 2000 +25=42025Similarly, 115^2: 100^2 + 2*100*15 +15^2=10000+3000+225=13225So, distance squared is (42025 + 13225)/ (38^2) = 55250 / 1444Simplify 55250 /1444. Let's see if we can reduce this fraction.Divide numerator and denominator by 2: 27625 /722Hmm, 722 is 2*361, and 361 is 19^2. 27625 divided by 19: 19*1450=27550, so 27625-27550=75. So, 27625=19*1450 +75. 75 is 19*3 + 18. So, it's not a multiple of 19. Therefore, the fraction is 27625/722.So, the radius squared is 27625/722.Now, the length of the tangent from the origin is sqrt[h^2 + k^2 - r^2], where h=357/38, k=-1/38, and r^2=27625/722.Compute h^2 + k^2:h^2 = (357/38)^2 = (357^2)/(38^2). 357^2: Let's compute 350^2 + 2*350*7 +7^2=122500 +4900 +49=127449So, h^2 =127449 /1444k^2 = (-1/38)^2 =1/1444So, h^2 + k^2 = (127449 +1)/1444=127450/1444Now, subtract r^2: 127450/1444 -27625/722But 27625/722 is equal to 55250/1444, since 722*2=1444 and 27625*2=55250.So, h^2 +k^2 -r^2=127450/1444 -55250/1444=(127450 -55250)/1444=72200/1444Simplify 72200/1444. Let's divide numerator and denominator by 4: 18050/361.Wait, 361*50=18050, so 18050/361=50.So, h^2 +k^2 -r^2=50.Therefore, the length of the tangent is sqrt(50)=5*sqrt(2). Wait, that can't be right because earlier I thought it was 10. Wait, let me check.Wait, 72200 divided by 1444: 1444*50=72200. So, yes, it's 50. So, sqrt(50)=5*sqrt(2). Hmm, but in the initial problem, the user mentioned getting 10, but my calculation shows 5*sqrt(2). Did I make a mistake?Wait, let me double-check the calculations.First, center at (357/38, -1/38). Let me confirm that.Midpoint of AB: (6, 4.5). Perpendicular bisector slope: -4/3. Equation: y = (-4/3)x +25/2.Midpoint of AC: (8.5,4). Perpendicular bisector slope: -9/2. Equation: y = (-9/2)x +169/4.Solving these two equations:(-4/3)x +25/2 = (-9/2)x +169/4Multiply by 12: -16x +150 = -54x +507So, 38x = 357 => x=357/38=9.3947.Then, y = (-4/3)(357/38) +25/2.Compute (-4/3)(357/38)= (-4*357)/(3*38)= (-1428)/114= (-1428 ÷6)/(114 ÷6)= (-238)/19= -12.5263.25/2=12.5. So, y= -12.5263 +12.5= -0.0263, which is -1/38. So, center is correct.Radius squared from center to A: (4 -357/38)^2 + (3 +1/38)^2.Compute 4=152/38, so 152/38 -357/38= -205/38.3=114/38, so 114/38 +1/38=115/38.So, squared distance: (205/38)^2 + (115/38)^2= (42025 +13225)/1444=55250/1444=27625/722.Then, h^2 +k^2= (357/38)^2 + (1/38)^2= (127449 +1)/1444=127450/1444.Subtract r^2: 127450/1444 -27625/722=127450/1444 -55250/1444=72200/1444=50.So, sqrt(50)=5*sqrt(2). So, the length is 5*sqrt(2). But the initial thought was 10. So, maybe I made a mistake in the calculation.Wait, let me check the power of a point formula. The power of the origin with respect to the circle is equal to the square of the tangent length. The power is also equal to (distance from origin to center)^2 - radius^2.Wait, yes, that's exactly what I computed: h^2 +k^2 -r^2=50, so tangent length is sqrt(50)=5*sqrt(2). So, why did the initial thought say 10? Maybe the initial thought was incorrect.Wait, let me verify with another method. Maybe using coordinates.Alternatively, the equation of the circle can be found using the general equation: x^2 + y^2 + Dx + Ey + F=0.Plugging in points A, B, C:For A(4,3): 16 +9 +4D +3E +F=0 =>25 +4D +3E +F=0For B(8,6):64 +36 +8D +6E +F=0 =>100 +8D +6E +F=0For C(13,5):169 +25 +13D +5E +F=0 =>194 +13D +5E +F=0So, we have three equations:1) 4D +3E +F = -252)8D +6E +F = -1003)13D +5E +F = -194Let me subtract equation 1 from equation 2:(8D -4D) + (6E -3E) + (F -F) = (-100 - (-25))4D +3E = -75Similarly, subtract equation 2 from equation 3:(13D -8D) + (5E -6E) + (F -F) = (-194 - (-100))5D -E = -94Now, we have two equations:4D +3E = -75 ...(a)5D -E = -94 ...(b)Let me solve equation (b) for E: E=5D +94.Plug into equation (a):4D +3*(5D +94)= -754D +15D +282= -7519D= -75 -282= -357So, D= -357/19= -18.789. Wait, 357 divided by 19: 19*18=342, 357-342=15, so D= -18 -15/19= -18.789.Then, E=5D +94=5*(-357/19)+94= (-1785/19)+94.Convert 94 to nineteenths: 94=1786/19.So, E= (-1785 +1786)/19=1/19.So, E=1/19.Now, from equation 1:4D +3E +F= -25.Plug D and E:4*(-357/19) +3*(1/19) +F= -25Compute 4*(-357/19)= (-1428)/193*(1/19)=3/19So, (-1428 +3)/19 +F= -25(-1425)/19 +F= -25-1425/19= -75So, -75 +F= -25 => F= -25 +75=50.So, the equation of the circle is x^2 + y^2 + Dx + Ey +F=0, which is x^2 + y^2 - (357/19)x + (1/19)y +50=0.Now, the center is (-D/2, -E/2)= (357/38, -1/38), which matches what I found earlier. The radius squared is (D/2)^2 + (E/2)^2 -F.Compute (357/38)^2 + (1/38)^2 -50.Which is (127449 +1)/1444 -50=127450/1444 -50.127450/1444= approx 88.25. 88.25 -50=38.25. Wait, but earlier I had radius squared as 27625/722≈38.25. So, that's consistent.Wait, but the power of the origin is h^2 +k^2 -r^2=50, so the tangent length is sqrt(50)=5*sqrt(2). So, that's correct.But in the initial problem, the user thought it was 10. So, maybe the user made a mistake. Alternatively, perhaps I made a miscalculation.Wait, let me check the power of the origin again. The power is |OT|^2= h^2 +k^2 -r^2=50, so |OT|=sqrt(50)=5*sqrt(2). So, that's correct.Alternatively, maybe I can compute the tangent length using coordinates.The tangent from the origin to the circle will satisfy the condition that the distance from the origin to the tangent line is equal to the radius. Wait, no, that's not the case. The tangent line from the origin touches the circle at one point, and the length of the tangent is sqrt(h^2 +k^2 -r^2).Alternatively, maybe I can parametrize the tangent line and find its intersection with the circle, but that seems more complicated.Alternatively, using coordinates, the tangent from the origin will satisfy the condition that the line from the origin to the point of tangency is perpendicular to the radius at that point. So, if T is the point of tangency, then OT is perpendicular to CT, where C is the center.So, the vector OT is perpendicular to the vector CT. So, their dot product is zero.Let me denote T as (x,y). Then, OT=(x,y), CT=(x - h, y -k). Their dot product is x(x - h) + y(y -k)=0.Also, since T lies on the circle, it satisfies (x - h)^2 + (y -k)^2 = r^2.So, we have two equations:1) x(x - h) + y(y -k)=02) (x - h)^2 + (y -k)^2 = r^2Subtracting equation 1 from equation 2:(x - h)^2 + (y -k)^2 - [x(x - h) + y(y -k)] = r^2 -0Expand (x - h)^2 =x^2 -2hx +h^2(y -k)^2=y^2 -2ky +k^2So, equation 2 - equation 1:(x^2 -2hx +h^2 + y^2 -2ky +k^2) - (x^2 -hx + y^2 -ky) = r^2Simplify:x^2 -2hx +h^2 + y^2 -2ky +k^2 -x^2 +hx -y^2 +ky = r^2Combine like terms:(-2hx +hx) + (-2ky +ky) + (h^2 +k^2) = r^2(-hx) + (-ky) + (h^2 +k^2) = r^2So, -hx -ky + h^2 +k^2 = r^2Which rearranges to:hx + ky = h^2 +k^2 - r^2But from the power of a point, we know that h^2 +k^2 - r^2 = |OT|^2, which is 50. So, hx + ky =50.But also, from equation 1: x(x - h) + y(y -k)=0 =>x^2 -hx + y^2 -ky=0 =>x^2 + y^2 =hx +ky.But hx +ky=50, so x^2 + y^2=50.So, the equation of the tangent line from the origin is x^2 + y^2=50. Wait, that's a circle itself. Hmm, no, actually, the tangent line from the origin to the circle will satisfy this condition, but it's a bit more involved.Alternatively, since we have the equation of the circle, we can find the equation of the tangent from the origin.The condition for a line y=mx to be tangent to the circle x^2 + y^2 + Dx + Ey +F=0 is that the distance from the center to the line equals the radius.Wait, but the line is y=mx, passing through the origin. The distance from the center (h,k) to the line y=mx is |mh -k| / sqrt(m^2 +1). This should equal the radius r.So, |mh -k| / sqrt(m^2 +1) = r.We can solve for m.Given that h=357/38, k=-1/38, r^2=27625/722, so r= sqrt(27625/722)=sqrt(27625)/sqrt(722)=166.25/26.87≈6.18.Wait, but let's compute it exactly.r^2=27625/722. So, r= sqrt(27625/722)=sqrt(27625)/sqrt(722). 27625=25*1105=25*5*221=25*5*13*17. So, sqrt(27625)=5*sqrt(1105). Similarly, sqrt(722)=sqrt(2*361)=sqrt(2)*19.So, r=5*sqrt(1105)/(19*sqrt(2))=5*sqrt(2210)/38.But this seems complicated. Alternatively, let's use the condition:|mh -k| / sqrt(m^2 +1) = rSo, |m*(357/38) - (-1/38)| / sqrt(m^2 +1) = sqrt(27625/722)Simplify numerator: |(357m +1)/38| / sqrt(m^2 +1) = sqrt(27625/722)Multiply both sides by 38:|357m +1| / sqrt(m^2 +1) = 38*sqrt(27625/722)=38*sqrt(27625)/sqrt(722)Compute sqrt(27625)=166.25, sqrt(722)=26.87. So, 38*166.25/26.87≈38*6.18≈235.But let's do it exactly:38*sqrt(27625/722)=38*sqrt(27625)/sqrt(722)=38*sqrt(27625)/sqrt(722)But 27625=25*1105, 722=2*361=2*19^2.So, sqrt(27625)=5*sqrt(1105), sqrt(722)=19*sqrt(2).Thus, 38*5*sqrt(1105)/(19*sqrt(2))= (38/19)*(5/sqrt(2))*sqrt(1105)=2*(5/sqrt(2))*sqrt(1105)=10*sqrt(1105/2)=10*sqrt(552.5).Wait, this seems messy. Maybe there's a better way.Alternatively, square both sides:(357m +1)^2 / (m^2 +1) = (27625/722)*(38^2)Compute 38^2=1444.So, RHS= (27625/722)*1444=27625*(1444/722)=27625*2=55250.So, (357m +1)^2 =55250*(m^2 +1)Expand LHS: (357m)^2 +2*357m*1 +1=127449m^2 +714m +1So, 127449m^2 +714m +1=55250m^2 +55250Bring all terms to left:127449m^2 -55250m^2 +714m +1 -55250=0Compute 127449-55250=72199So, 72199m^2 +714m -55249=0This is a quadratic in m: 72199m^2 +714m -55249=0This seems very messy. Maybe I made a mistake earlier.Alternatively, since we know the power of the origin is 50, the tangent length is sqrt(50)=5*sqrt(2). So, maybe I should just accept that as the answer.But wait, why did the initial thought say 10? Maybe the user made a mistake in their calculation. Alternatively, perhaps I made a mistake in computing h^2 +k^2 -r^2.Wait, let me recompute h^2 +k^2 -r^2.h=357/38, so h^2= (357)^2/(38)^2=127449/1444k=-1/38, so k^2=1/1444r^2=27625/722=55250/1444So, h^2 +k^2 -r^2= (127449 +1 -55250)/1444= (127450 -55250)/1444=72200/1444=50.Yes, that's correct. So, the tangent length is sqrt(50)=5*sqrt(2).Therefore, the length is 5*sqrt(2). So, the initial thought of 10 was incorrect.Wait, but let me check with another method. Maybe using coordinates and solving for the tangent line.The equation of the circle is x^2 + y^2 - (357/19)x + (1/19)y +50=0.The tangent from the origin will satisfy the condition that the distance from the origin to the tangent line equals the radius.Wait, no, the tangent line from the origin touches the circle at one point, and the distance from the origin to the center minus the radius gives the tangent length.Wait, no, the tangent length is sqrt(h^2 +k^2 -r^2)=sqrt(50)=5*sqrt(2). So, that's correct.Alternatively, using coordinates, the tangent points can be found by solving the system:x^2 + y^2 - (357/19)x + (1/19)y +50=0andy = mxSubstitute y=mx into the circle equation:x^2 + (mx)^2 - (357/19)x + (1/19)(mx) +50=0(1 +m^2)x^2 + (-357/19 +m/19)x +50=0For this to have exactly one solution, the discriminant must be zero.Discriminant D= [(-357/19 +m/19)]^2 -4*(1 +m^2)*50=0Compute:Let me write it as:[(m -357)/19]^2 -200*(1 +m^2)=0Expand [(m -357)/19]^2= (m^2 -714m +127449)/361So, equation becomes:(m^2 -714m +127449)/361 -200 -200m^2=0Multiply all terms by 361 to eliminate denominator:m^2 -714m +127449 -200*361 -200m^2*361=0Compute 200*361=72200200m^2*361=72200m^2So, equation becomes:m^2 -714m +127449 -72200 -72200m^2=0Combine like terms:(1 -72200)m^2 -714m + (127449 -72200)=0-72199m^2 -714m +55249=0Multiply both sides by -1:72199m^2 +714m -55249=0Which is the same quadratic as before. So, solving this quadratic will give the slopes of the tangent lines.But solving 72199m^2 +714m -55249=0 is quite involved. Let me use the quadratic formula:m = [-714 ± sqrt(714^2 -4*72199*(-55249))]/(2*72199)Compute discriminant:D=714^2 +4*72199*55249714^2=510,000 approximately. Wait, 700^2=490,000, 14^2=196, so 714^2= (700+14)^2=700^2 +2*700*14 +14^2=490000 +19600 +196=509,796.4*72199*55249: Let's compute 4*72199=288,796. Then, 288,796*55,249. This is a huge number. Let me see if I can factor something out.Alternatively, perhaps I can factor the quadratic equation.But given the size of the coefficients, it's impractical. So, perhaps it's better to accept that the tangent length is 5*sqrt(2).Therefore, the length of the tangent from the origin to the circle is 5*sqrt(2).

💡Okay, so I have this problem about an arithmetic sequence. It says that in the sequence {a_n}, the sum of the second term and the fourth term is 6. I need to find the sum of the first five terms of this sequence. Hmm, let me think about how to approach this.First, I remember that in an arithmetic sequence, each term is obtained by adding a constant difference to the previous term. So, if I denote the first term as a_1 and the common difference as d, then the nth term can be written as a_n = a_1 + (n-1)d. That seems right.Given that, let me write down the expressions for a_2 and a_4. a_2 would be the second term, so that's a_1 + (2-1)d = a_1 + d.Similarly, a_4 is the fourth term, which is a_1 + (4-1)d = a_1 + 3d.The problem states that a_2 + a_4 = 6. So, substituting the expressions I just wrote:(a_1 + d) + (a_1 + 3d) = 6.Let me simplify that:a_1 + d + a_1 + 3d = 6.Combine like terms:2a_1 + 4d = 6.Hmm, okay, so I have this equation: 2a_1 + 4d = 6. I can simplify this further by dividing both sides by 2:a_1 + 2d = 3.Alright, so that's one equation. Now, I need to find the sum of the first five terms. The formula for the sum of the first n terms of an arithmetic sequence is S_n = n/2 * (2a_1 + (n-1)d). So, for n=5, it would be:S_5 = 5/2 * (2a_1 + 4d).Wait a second, looking at the expression inside the parentheses: 2a_1 + 4d. From my earlier equation, I know that 2a_1 + 4d = 6. So, that simplifies things.So, substituting back into the sum formula:S_5 = 5/2 * 6.Calculating that:5/2 * 6 = (5 * 6)/2 = 30/2 = 15.So, the sum of the first five terms is 15.But let me double-check to make sure I didn't make any mistakes. Starting from the beginning:Given a_2 + a_4 = 6.Expressed in terms of a_1 and d:(a_1 + d) + (a_1 + 3d) = 6.Simplify:2a_1 + 4d = 6.Divide by 2:a_1 + 2d = 3.Now, the sum of the first five terms is S_5 = 5/2 * (2a_1 + 4d).But 2a_1 + 4d is exactly the same as 2*(a_1 + 2d), which is 2*3 = 6.So, S_5 = 5/2 * 6 = 15.Yep, that checks out. So, the sum is indeed 15.