💡Okay, so I have this hyperbola equation: (frac{x^2}{a^2} - frac{y^2}{b^2} = 1), where (a > 0) and (b > 0). I need to find its eccentricity. Hmm, eccentricity is something I remember relates to how "stretched" the hyperbola is. The formula for eccentricity (e) is (e = frac{c}{a}), where (c) is the distance from the center to each focus. I also recall that for hyperbolas, (c^2 = a^2 + b^2). So, if I can find the relationship between (a) and (b), I can find (e).The problem mentions the left focus and a vertical line to the (x)-axis intersecting the hyperbola at points (M) and (N). Then, a circle with diameter (MN) passes through the right vertex of the hyperbola. Let me try to visualize this.First, the hyperbola is centered at the origin, opening left and right. The left focus is at ((-c, 0)), and the right vertex is at ((a, 0)). The vertical line to the (x)-axis... Wait, vertical lines are parallel to the (y)-axis, so a vertical line would have the equation (x = k) for some constant (k). Since it's intersecting the hyperbola, this line must cross the hyperbola at two points, (M) and (N).But the problem says the vertical line is "vertical to the (x)-axis." Wait, that might be a translation issue. Maybe it means a line perpendicular to the (x)-axis, which would indeed be a vertical line. So, the vertical line passes through the left focus, which is at ((-c, 0)). Therefore, the equation of this vertical line is (x = -c).So, this line (x = -c) intersects the hyperbola at points (M) and (N). Let me find the coordinates of these points. Substitute (x = -c) into the hyperbola equation:[frac{(-c)^2}{a^2} - frac{y^2}{b^2} = 1]Simplify:[frac{c^2}{a^2} - frac{y^2}{b^2} = 1]I know that (c^2 = a^2 + b^2), so substitute that in:[frac{a^2 + b^2}{a^2} - frac{y^2}{b^2} = 1]Simplify the first term:[1 + frac{b^2}{a^2} - frac{y^2}{b^2} = 1]Subtract 1 from both sides:[frac{b^2}{a^2} - frac{y^2}{b^2} = 0]Multiply both sides by (a^2 b^2) to eliminate denominators:[b^4 - a^2 y^2 = 0]So,[a^2 y^2 = b^4]Divide both sides by (a^2):[y^2 = frac{b^4}{a^2}]Take square roots:[y = pm frac{b^2}{a}]Therefore, the points (M) and (N) are ((-c, frac{b^2}{a})) and ((-c, -frac{b^2}{a})).Now, the circle with diameter (MN) passes through the right vertex ((a, 0)). Let me recall that the equation of a circle with diameter endpoints ((x_1, y_1)) and ((x_2, y_2)) is:[(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0]So, plugging in (M(-c, frac{b^2}{a})) and (N(-c, -frac{b^2}{a})):[(x - (-c))(x - (-c)) + (y - frac{b^2}{a})(y - (-frac{b^2}{a})) = 0]Simplify:[(x + c)^2 + left(y - frac{b^2}{a}right)left(y + frac{b^2}{a}right) = 0]Multiply out the terms:First, expand ((x + c)^2):[x^2 + 2cx + c^2]Then, the product of the (y) terms is a difference of squares:[y^2 - left(frac{b^2}{a}right)^2]So, putting it all together:[x^2 + 2cx + c^2 + y^2 - frac{b^4}{a^2} = 0]Combine like terms:[x^2 + y^2 + 2cx + c^2 - frac{b^4}{a^2} = 0]Now, since the circle passes through the right vertex ((a, 0)), substitute (x = a) and (y = 0) into the equation:[a^2 + 0 + 2c(a) + c^2 - frac{b^4}{a^2} = 0]Simplify:[a^2 + 2ac + c^2 - frac{b^4}{a^2} = 0]Let me write this as:[(a + c)^2 - frac{b^4}{a^2} = 0]So,[(a + c)^2 = frac{b^4}{a^2}]Take square roots on both sides (though I need to be careful with signs, but since all terms are positive, it's fine):[a + c = frac{b^2}{a}]Multiply both sides by (a):[a^2 + a c = b^2]But I also know that for hyperbolas, (c^2 = a^2 + b^2). So, substitute (b^2) from the previous equation:[c^2 = a^2 + (a^2 + a c)]Simplify:[c^2 = 2a^2 + a c]Bring all terms to one side:[c^2 - a c - 2a^2 = 0]This is a quadratic equation in terms of (c). Let me write it as:[c^2 - a c - 2a^2 = 0]Let me solve for (c) using the quadratic formula. Let me consider (c) as the variable and (a) as a constant. So, the quadratic is (c^2 - a c - 2a^2 = 0). The quadratic formula is (c = frac{-b pm sqrt{b^2 - 4ac}}{2a}), but here the coefficients are:- Coefficient of (c^2): 1- Coefficient of (c): -a- Constant term: -2a^2So, applying the quadratic formula:[c = frac{a pm sqrt{a^2 + 8a^2}}{2}]Simplify under the square root:[sqrt{9a^2} = 3a]So,[c = frac{a pm 3a}{2}]This gives two solutions:1. (c = frac{a + 3a}{2} = frac{4a}{2} = 2a)2. (c = frac{a - 3a}{2} = frac{-2a}{2} = -a)But (c) is a distance from the center to the focus, so it must be positive. Since (a > 0), (c = 2a) is the valid solution.Therefore, (c = 2a). The eccentricity (e) is (e = frac{c}{a} = frac{2a}{a} = 2).Wait, let me double-check my steps. When I had (a + c = frac{b^2}{a}), I substituted (b^2 = a^2 + a c) into (c^2 = a^2 + b^2), which gave me (c^2 = 2a^2 + a c). Then, rearranged to (c^2 - a c - 2a^2 = 0), solved quadratic, got (c = 2a). That seems consistent.So, yes, the eccentricity is 2.**Final Answer**The eccentricity of the hyperbola is (boxed{2}).

💡Okay, so I have this problem here about sequences. Let me try to understand what it's asking. It says that the sequence {log₂(aₙ - 1)} is an arithmetic sequence, and we're given that a₁ = 3 and a₂ = 5. There are two parts to the problem: first, I need to prove that the sequence {aₙ - 1} is a geometric sequence, and second, I have to calculate the value of this sum: 1/(a₂ - a₁) + 1/(a₃ - a₂) + ... + 1/(a_{n+1} - aₙ).Alright, starting with part (1). So, we know that {log₂(aₙ - 1)} is arithmetic. An arithmetic sequence has a common difference between consecutive terms. That means log₂(a₂ - 1) - log₂(a₁ - 1) should be equal to log₂(a₃ - 1) - log₂(a₂ - 1), and so on. Let me compute that difference first.Given a₁ = 3, so a₁ - 1 = 2. Similarly, a₂ = 5, so a₂ - 1 = 4. Therefore, log₂(a₁ - 1) = log₂(2) = 1, and log₂(a₂ - 1) = log₂(4) = 2. So the difference between the first two terms is 2 - 1 = 1. Let me check if this difference is consistent for the next term. Wait, but I don't have a₃ yet. Maybe I can find a₃?Since {log₂(aₙ - 1)} is arithmetic with common difference 1, the next term should be log₂(a₃ - 1) = log₂(a₂ - 1) + 1 = 2 + 1 = 3. Therefore, a₃ - 1 = 2³ = 8, so a₃ = 9. Let me verify if this makes sense.So, a₁ = 3, a₂ = 5, a₃ = 9. Let's see the differences: a₂ - a₁ = 2, a₃ - a₂ = 4. Hmm, those differences are 2 and 4. That seems like a geometric progression with a common ratio of 2. So, {aₙ - 1} is 2, 4, 8,... which is clearly a geometric sequence with first term 2 and common ratio 2.Wait, but the problem says to prove that {aₙ - 1} is a geometric sequence. So, maybe I can generalize this. Since {log₂(aₙ - 1)} is arithmetic, that means log₂(aₙ - 1) = log₂(a₁ - 1) + (n - 1)d, where d is the common difference. We already found that d = 1 because log₂(4) - log₂(2) = 1.So, log₂(aₙ - 1) = 1 + (n - 1)*1 = n. Therefore, aₙ - 1 = 2ⁿ. So, {aₙ - 1} is indeed a geometric sequence with first term 2 and common ratio 2. That proves part (1).Moving on to part (2). I need to calculate the sum S = 1/(a₂ - a₁) + 1/(a₃ - a₂) + ... + 1/(a_{n+1} - aₙ). From part (1), we know that aₙ - 1 = 2ⁿ, so aₙ = 2ⁿ + 1. Therefore, a_{n+1} = 2^{n+1} + 1.So, the difference a_{n+1} - aₙ = (2^{n+1} + 1) - (2ⁿ + 1) = 2^{n+1} - 2ⁿ = 2ⁿ(2 - 1) = 2ⁿ. Therefore, 1/(a_{n+1} - aₙ) = 1/2ⁿ.So, the sum S becomes 1/2 + 1/2² + 1/2³ + ... + 1/2ⁿ. That's a finite geometric series with first term 1/2 and common ratio 1/2, summed up to n terms.The formula for the sum of a geometric series is S = a(1 - rⁿ)/(1 - r), where a is the first term, r is the common ratio, and n is the number of terms. Plugging in the values, we get S = (1/2)(1 - (1/2)ⁿ)/(1 - 1/2) = (1/2)(1 - 1/2ⁿ)/(1/2) = (1 - 1/2ⁿ).So, the sum simplifies to 1 - 1/2ⁿ.Wait, let me double-check that. The first term is 1/2, the ratio is 1/2, and the number of terms is n. So, applying the formula:Sum = (first term) * (1 - ratioⁿ)/(1 - ratio) = (1/2)*(1 - (1/2)ⁿ)/(1 - 1/2) = (1/2)*(1 - 1/2ⁿ)/(1/2) = (1 - 1/2ⁿ). Yep, that looks correct.So, the value of the sum is 1 - 1/2ⁿ.I think that's it. I've gone through the steps, checked my calculations, and everything seems consistent. I didn't make any mistakes that I can see, so I'm confident in my answers.**Final Answer**(1) boxed{a_n - 1} is a geometric sequence.(2) The value of the sum is boxed{1 - frac{1}{2^n}}.

💡First, I need to determine the volume of water in the garage. The garage has a rectangular floor measuring 20 feet by 25 feet, and the water depth is 12 inches. Since the depth is given in inches, I'll convert it to feet for consistency.Next, I'll calculate the volume of water by multiplying the length, width, and depth of the flooded area.Once I have the volume in cubic feet, I'll convert it to gallons using the conversion factor that 1 cubic foot of water equals 7.5 gallons.After finding the total number of gallons, I'll consider the pumping rate. There are two pumps, each capable of pumping 10 gallons per minute. I'll calculate the combined pumping rate of both pumps.Finally, I'll determine the time required to pump out all the water by dividing the total gallons of water by the total pumping rate. This will give me the time in minutes needed to remove all the water from the garage.

💡First, I need to simplify the expression (2 - frac{2}{1 + sqrt{2}} - frac{2}{1 - sqrt{2}}). I notice that the denominators of the fractions contain radicals, so I should consider rationalizing them to make the expression easier to work with.To rationalize (frac{2}{1 + sqrt{2}}), I multiply the numerator and the denominator by the conjugate of the denominator, which is (1 - sqrt{2}):[frac{2}{1 + sqrt{2}} times frac{1 - sqrt{2}}{1 - sqrt{2}} = frac{2(1 - sqrt{2})}{(1 + sqrt{2})(1 - sqrt{2})}]Simplifying the denominator using the difference of squares:[(1 + sqrt{2})(1 - sqrt{2}) = 1 - (sqrt{2})^2 = 1 - 2 = -1]So, the fraction becomes:[frac{2(1 - sqrt{2})}{-1} = -2(1 - sqrt{2}) = -2 + 2sqrt{2}]Next, I rationalize (frac{2}{1 - sqrt{2}}) by multiplying the numerator and the denominator by the conjugate (1 + sqrt{2}):[frac{2}{1 - sqrt{2}} times frac{1 + sqrt{2}}{1 + sqrt{2}} = frac{2(1 + sqrt{2})}{(1 - sqrt{2})(1 + sqrt{2})}]Again, simplifying the denominator:[(1 - sqrt{2})(1 + sqrt{2}) = 1 - (sqrt{2})^2 = 1 - 2 = -1]So, the fraction becomes:[frac{2(1 + sqrt{2})}{-1} = -2(1 + sqrt{2}) = -2 - 2sqrt{2}]Now, I substitute the simplified fractions back into the original expression:[2 - frac{2}{1 + sqrt{2}} - frac{2}{1 - sqrt{2}} = 2 - (-2 + 2sqrt{2}) - (-2 - 2sqrt{2})]Simplifying the expression by removing the parentheses:[2 + 2 - 2sqrt{2} + 2 + 2sqrt{2}]Combining like terms:[2 + 2 + 2 = 6]And the terms with (sqrt{2}) cancel out:[-2sqrt{2} + 2sqrt{2} = 0]So, the simplified expression is:[6]However, this doesn't match any of the provided options. I must have made a mistake in my calculations. Let me check my steps again.Upon reviewing, I realize that when I substituted the simplified fractions back into the expression, I incorrectly applied the negative signs. The correct substitution should be:[2 - (-2 + 2sqrt{2}) - (-2 - 2sqrt{2}) = 2 + 2 - 2sqrt{2} + 2 + 2sqrt{2}]Combining like terms:[2 + 2 + 2 = 6]And the terms with (sqrt{2}) cancel out:[-2sqrt{2} + 2sqrt{2} = 0]So, the simplified expression is indeed:[6]But since this doesn't match any of the options, I must have made an error in my initial approach. Perhaps there's a different method to simplify the expression.Let me try another approach by combining the two fractions first before simplifying:[frac{2}{1 + sqrt{2}} + frac{2}{1 - sqrt{2}} = 2left(frac{1}{1 + sqrt{2}} + frac{1}{1 - sqrt{2}}right)]Combining the fractions:[2left(frac{1 - sqrt{2} + 1 + sqrt{2}}{(1 + sqrt{2})(1 - sqrt{2})}right) = 2left(frac{2}{-1}right) = 2(-2) = -4]Now, substituting back into the original expression:[2 - (-4) = 2 + 4 = 6]Again, I arrive at the same result, which doesn't match any of the provided options. It seems there's a discrepancy between my calculations and the answer choices. Perhaps I misinterpreted the original expression or there's a typo in the problem.Given the options provided, the closest match to my calculations is option D) (-2), but my result is (6). This suggests that there might be an error in the problem statement or the answer choices. However, based on the standard simplification process, the correct answer should be (6), which isn't listed among the options. Therefore, I might need to revisit the problem or check if there's a different interpretation.

💡First, I need to determine the principal amount using the simple interest formula. I know the simple interest, rate, and time.I'll start by rearranging the simple interest formula to solve for the principal.Next, I'll substitute the given values into the formula.Then, I'll perform the multiplication in the denominator.After that, I'll divide the simple interest by the result to find the principal.Finally, I'll present the calculated principal as the solution.

💡Okay, so I have this geometry problem about a cube and a sphere inside it. The sphere has a radius of 10 and is centered at point O. The sphere intersects three different faces of the cube, creating circles with radii 1, 1, and 3. I need to find the length of the segment OD₁, which is the distance from the center of the sphere O to the vertex D₁ of the cube.First, I need to visualize the cube and the sphere inside it. The cube has vertices labeled ABCD A₁B₁C₁D₁, which means it's a standard cube with edges connecting corresponding vertices. The sphere is inside this cube, so the center O is somewhere within the cube.The sphere intersects three faces: AA₁D₁D, A₁B₁C₁D₁, and CDD₁C₁. Each intersection creates a circle. The radii of these circles are 1, 1, and 3 respectively. I remember that when a sphere intersects a plane, the intersection is a circle whose radius can be found using the Pythagorean theorem if we know the distance from the sphere's center to the plane.Let me denote the distance from O to each of these planes as d₁, d₂, and d₃. Since the sphere has a radius of 10, the relationship between the radius of the circle (r) and the distance from the center to the plane (d) is given by:r² + d² = R²Where R is the radius of the sphere. So, for each intersection:1. For the face AA₁D₁D with radius 1: 1² + d₁² = 10² 1 + d₁² = 100 d₁² = 99 d₁ = √992. For the face A₁B₁C₁D₁ with radius 1: 1² + d₂² = 10² 1 + d₂² = 100 d₂² = 99 d₂ = √993. For the face CDD₁C₁ with radius 3: 3² + d₃² = 10² 9 + d₃² = 100 d₃² = 91 d₃ = √91So, the distances from O to these three planes are √99, √99, and √91.Now, I need to relate these distances to the coordinates of O within the cube. Let me assume that the cube has edge length 'a'. Since the sphere is inside the cube, the edge length must be at least twice the radius of the sphere, but since the sphere intersects the faces, the edge length is actually larger. However, I don't know the edge length yet, so maybe I can express the coordinates of O in terms of the cube's dimensions.Let me set up a coordinate system with vertex A at the origin (0,0,0). Then, the cube can be defined with vertices:- A(0,0,0)- B(a,0,0)- C(a,a,0)- D(0,a,0)- A₁(0,0,a)- B₁(a,0,a)- C₁(a,a,a)- D₁(0,a,a)So, point D₁ is at (0,a,a). The center O is somewhere inside the cube, so its coordinates are (x,y,z), where 0 < x < a, 0 < y < a, 0 < z < a.Now, the distance from O to each face can be related to its coordinates. The distance from O to face AA₁D₁D (which is the face on the y-z plane at x=0) is simply x. Similarly, the distance to face A₁B₁C₁D₁ (which is the top face at z=a) is a - z. The distance to face CDD₁C₁ (which is the face on the x-z plane at y=a) is a - y.Wait, hold on. Let me verify that. The distance from a point (x,y,z) to the face AA₁D₁D (x=0) is indeed x. The distance to face A₁B₁C₁D₁ (z=a) is a - z. The distance to face CDD₁C₁ (y=a) is a - y.But earlier, I found the distances d₁, d₂, d₃ as √99, √99, and √91. So, mapping these:- d₁ = x = √99- d₂ = a - z = √99- d₃ = a - y = √91So, from these, I can express x, y, z in terms of a:x = √99a - z = √99 ⇒ z = a - √99a - y = √91 ⇒ y = a - √91So, the coordinates of O are (√99, a - √91, a - √99).Now, I need to find the length of OD₁. Point D₁ is at (0,a,a). So, the distance between O(√99, a - √91, a - √99) and D₁(0,a,a) can be found using the distance formula in 3D:OD₁ = √[(√99 - 0)² + (a - √91 - a)² + (a - √99 - a)²]Simplify each term:First term: (√99)² = 99Second term: (a - √91 - a)² = (-√91)² = 91Third term: (a - √99 - a)² = (-√99)² = 99So, OD₁ = √[99 + 91 + 99] = √[289] = 17Wait, that seems straightforward. But let me double-check my reasoning.I assumed that the distances from O to the respective faces are equal to x, a - z, and a - y. Is that correct?Yes, because in a cube, the distance from a point inside to a face is the coordinate perpendicular to that face. For example, the distance to the face x=0 is x, the distance to the face z=a is a - z, and the distance to the face y=a is a - y.So, my expressions for d₁, d₂, d₃ in terms of x, y, z are correct.Then, I expressed x, y, z in terms of a:x = √99z = a - √99y = a - √91So, O is at (√99, a - √91, a - √99)Then, D₁ is at (0,a,a). So, the differences in coordinates are:Δx = √99 - 0 = √99Δy = (a - √91) - a = -√91Δz = (a - √99) - a = -√99Then, squaring each difference:(√99)² = 99(-√91)² = 91(-√99)² = 99Adding them up: 99 + 91 + 99 = 289Square root of 289 is 17.Therefore, OD₁ = 17.I think that's correct. Let me just make sure I didn't make any calculation errors.Calculating the distances:For face AA₁D₁D: radius 1, so distance from O is √(10² - 1²) = √99For face A₁B₁C₁D₁: radius 1, same distance √99For face CDD₁C₁: radius 3, so distance from O is √(10² - 3²) = √91Then, using these distances as the coordinates relative to the cube's edges, and then computing the distance from O to D₁, which is at (0,a,a). The differences in coordinates squared give 99, 91, and 99, summing to 289, whose square root is 17.Yes, that seems consistent. I don't see any mistakes in the reasoning or calculations.**Final Answer**The length of the segment ( OD_1 ) is boxed{17}.