answer：Okay, so I have this problem about related curves, which are an ellipse and a hyperbola sharing the same foci and having reciprocal eccentricities. The goal is to find the eccentricity of the ellipse when the angle between the two foci at their intersection point P is 60 degrees.First, let me recall some basic properties of ellipses and hyperbolas. Both ellipses and hyperbolas have two foci, and for an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant, equal to the major axis length, which is 2a. For a hyperbola, the difference of the distances from any point on the hyperbola to the two foci is constant, equal to the transverse axis length, which is 2a'.Given that the ellipse and hyperbola share the same foci, let's denote the distance between the two foci as 2c. So, for both curves, the distance from the center to each focus is c.The eccentricity of an ellipse is given by e1 = c/a, and for the hyperbola, it's e2 = c/a'. The problem states that these eccentricities are reciprocals, so e1 * e2 = 1. That means (c/a) * (c/a') = 1, which simplifies to c^2 = a * a'.Now, the point P is an intersection point of the ellipse and hyperbola in the first quadrant. So, P lies on both curves, meaning it satisfies both the ellipse equation and the hyperbola equation.Let me denote the distances from P to F1 and F2 as m and n, respectively. So, for the ellipse, we have m + n = 2a, and for the hyperbola, we have |m - n| = 2a'. Since P is in the first quadrant, I think m and n are both positive, and without loss of generality, let's assume m > n, so m - n = 2a'.Given that the angle F1PF2 is 60 degrees, I can use the Law of Cosines in triangle F1PF2. The sides of the triangle are m, n, and 2c (the distance between F1 and F2). So, applying the Law of Cosines:(2c)^2 = m^2 + n^2 - 2mn cos(60°)Since cos(60°) is 0.5, this simplifies to:4c^2 = m^2 + n^2 - mnNow, from the ellipse and hyperbola definitions, we have:m + n = 2a ...(1)m - n = 2a' ...(2)If I solve these two equations for m and n, I can express m and n in terms of a and a'. Adding equations (1) and (2):2m = 2a + 2a' => m = a + a'Subtracting equation (2) from equation (1):2n = 2a - 2a' => n = a - a'So, m = a + a' and n = a - a'Now, let's substitute these into the equation from the Law of Cosines:4c^2 = (a + a')^2 + (a - a')^2 - (a + a')(a - a')Let me compute each term step by step.First, (a + a')^2 = a^2 + 2aa' + a'^2Second, (a - a')^2 = a^2 - 2aa' + a'^2Third, (a + a')(a - a') = a^2 - a'^2So, substituting back:4c^2 = [a^2 + 2aa' + a'^2] + [a^2 - 2aa' + a'^2] - [a^2 - a'^2]Let me simplify term by term.First, add the first two brackets:(a^2 + 2aa' + a'^2) + (a^2 - 2aa' + a'^2) = 2a^2 + 2a'^2Then subtract the third term:2a^2 + 2a'^2 - (a^2 - a'^2) = 2a^2 + 2a'^2 - a^2 + a'^2 = a^2 + 3a'^2So, 4c^2 = a^2 + 3a'^2But earlier, we had from the reciprocal eccentricities:c^2 = a * a'So, let's substitute c^2 in the equation:4(a * a') = a^2 + 3a'^2So, 4aa' = a^2 + 3a'^2Let me rearrange this equation:a^2 - 4aa' + 3a'^2 = 0This is a quadratic equation in terms of a. Let me write it as:a^2 - 4a a' + 3a'^2 = 0Let me factor this quadratic:Looking for two numbers that multiply to 3a'^2 and add to -4a'The factors would be -a' and -3a', because (-a')*(-3a') = 3a'^2 and (-a') + (-3a') = -4a'So, (a - a')(a - 3a') = 0Thus, either a = a' or a = 3a'But since a is the semi-major axis of the ellipse and a' is the semi-transverse axis of the hyperbola, and given that for an ellipse, a > c, and for a hyperbola, a' < c (since for hyperbola, c > a'), so a cannot be equal to a' because that would imply c^2 = a * a' = a^2, so c = a, which would make the ellipse a line segment, which is degenerate. So, a cannot equal a'.Therefore, the other solution is a = 3a'So, a = 3a'Now, since c^2 = a * a', and a = 3a', then c^2 = 3a' * a' = 3a'^2So, c = sqrt(3) a'Now, let's find the eccentricities.For the ellipse, e1 = c / aSince a = 3a', and c = sqrt(3) a', then:e1 = (sqrt(3) a') / (3a') = sqrt(3)/3For the hyperbola, e2 = c / a'Which is (sqrt(3) a') / a' = sqrt(3)And indeed, e1 * e2 = (sqrt(3)/3) * sqrt(3) = (3)/3 = 1, which satisfies the reciprocal condition.Therefore, the eccentricity of the ellipse is sqrt(3)/3.I think that's the answer. Let me just double-check my steps.1. Defined m and n as distances from P to F1 and F2.2. Used the Law of Cosines correctly with angle 60 degrees.3. Expressed m and n in terms of a and a' correctly.4. Substituted into the equation and expanded correctly.5. Factored the quadratic correctly, leading to a = 3a'6. Substituted back to find c in terms of a'7. Calculated e1 correctly as sqrt(3)/3Everything seems to check out. So, I'm confident that the eccentricity of the ellipse is sqrt(3)/3.

answer：Okay, so I have this problem where three friends—Amy, Jan, and Toy—are redistributing their money in a series of steps. Toy starts with 48 and ends with 48. I need to find the total amount of money all three friends have together. The answer choices are 144, 192, 240, 336, and 384, but I noticed that in the problem statement, there's a duplicate with 528 also listed. Hmm, maybe that's a typo, but I'll keep it in mind.Let me break down the problem step by step.First, let's define the initial amounts of money each person has. Let's say Amy has "a" dollars, Jan has "j" dollars, and Toy has "t" dollars. We know that Toy starts with 48, so t = 48.Now, the redistribution happens in three steps:1. **Amy gives money to Jan and Toy to triple their initial amounts.**2. **Jan then gives money to Amy and Toy to double their amounts after Amy's redistribution.**3. **Finally, Toy gives money to Amy and Jan to double their amounts after Jan's redistribution.**After all these transactions, Toy ends up with 48 again. So, we need to figure out how much money each person has after each step and set up equations accordingly.Let me tackle each step one by one.**Step 1: Amy gives money to Jan and Toy to triple their initial amounts.**So, initially, Jan has j dollars and Toy has t = 48 dollars. After Amy gives them money, Jan will have 3j and Toy will have 3t. That means Amy gives Jan 2j dollars and Toy 2t dollars. So, Amy's new amount will be a - 2j - 2t.Let me write that down:- Amy's amount after Step 1: a - 2j - 2t- Jan's amount after Step 1: 3j- Toy's amount after Step 1: 3t = 3*48 = 144**Step 2: Jan gives money to Amy and Toy to double their amounts after Step 1.**After Step 1, Amy has (a - 2j - 2t) and Toy has 144. Jan is going to give them money so that Amy and Toy each double their amounts.So, Amy's amount after Step 2 will be 2*(a - 2j - 2t), and Toy's amount after Step 2 will be 2*144 = 288.To double Amy's amount, Jan needs to give her (a - 2j - 2t) dollars. Similarly, to double Toy's amount, Jan needs to give her 144 dollars.Therefore, Jan's amount after Step 2 will be:3j - (a - 2j - 2t) - 144Let me simplify that:3j - a + 2j + 2t - 144 = (3j + 2j) - a + 2t - 144 = 5j - a + 2t - 144So, after Step 2:- Amy: 2*(a - 2j - 2t) = 2a - 4j - 4t- Jan: 5j - a + 2t - 144- Toy: 288**Step 3: Toy gives money to Amy and Jan to double their amounts after Step 2.**After Step 2, Amy has (2a - 4j - 4t) and Jan has (5j - a + 2t - 144). Toy is going to give them money so that Amy and Jan each double their amounts.So, Amy's amount after Step 3 will be 2*(2a - 4j - 4t) = 4a - 8j - 8tSimilarly, Jan's amount after Step 3 will be 2*(5j - a + 2t - 144) = 10j - 2a + 4t - 288To achieve this, Toy needs to give Amy (2a - 4j - 4t) dollars and Jan (5j - a + 2t - 144) dollars.Therefore, Toy's amount after Step 3 will be:288 - (2a - 4j - 4t) - (5j - a + 2t - 144)Let me simplify that:288 - 2a + 4j + 4t - 5j + a - 2t + 144Combine like terms:(288 + 144) + (-2a + a) + (4j - 5j) + (4t - 2t) = 432 - a - j + 2tWe know that after Step 3, Toy ends up with 48. So:432 - a - j + 2t = 48We can plug in t = 48:432 - a - j + 2*48 = 48Calculate 2*48 = 96:432 - a - j + 96 = 48Combine constants:432 + 96 = 528So:528 - a - j = 48Subtract 48 from both sides:528 - 48 - a - j = 0480 - a - j = 0Which means:a + j = 480So, the sum of Amy's and Jan's initial amounts is 480.But we need the total amount all three friends have, which is a + j + t.We know t = 48, so total = a + j + 48 = 480 + 48 = 528.Wait, 528 is one of the options, but it's not in the first list. It's in the second duplicate list. Maybe the problem had a typo, but 528 seems to be the correct total.Let me double-check my calculations to make sure I didn't make a mistake.Starting from the equation after Step 3:432 - a - j + 2t = 48t = 48, so:432 - a - j + 96 = 48432 + 96 = 528528 - a - j = 48So, 528 - 48 = a + j480 = a + jTotal = a + j + t = 480 + 48 = 528Yes, that seems correct. So, the total amount is 528.

answer：Alright, so I've got this problem about vectors. It says that vectors a and b satisfy |a + b| equals the square root of 10, and |a - b| equals the square root of 6. I need to find the dot product of a and b, which is written as a · b. The options are A. 1, B. 2, C. 3, D. 5.Okay, let's start by recalling what the magnitude of a vector means. The magnitude of a vector is like its length. So, |a + b| is the length of the vector when you add a and b together, and |a - b| is the length when you subtract b from a.I remember that when you have the magnitude of a vector, you can square it to make calculations easier. So, maybe I should square both sides of these equations. Let me try that.First, for |a + b| = sqrt(10), if I square both sides, I get |a + b|² = (sqrt(10))², which simplifies to |a + b|² = 10.Similarly, for |a - b| = sqrt(6), squaring both sides gives |a - b|² = (sqrt(6))², so |a - b|² = 6.Now, I need to figure out what |a + b|² and |a - b|² mean in terms of the dot product. I recall that the square of the magnitude of a vector can be expressed using the dot product. Specifically, |v|² = v · v for any vector v.So, applying that to |a + b|², it should be equal to (a + b) · (a + b). Let me write that out:|a + b|² = (a + b) · (a + b)Expanding this, I get:a · a + a · b + b · a + b · bSince the dot product is commutative, meaning a · b = b · a, this simplifies to:a · a + 2(a · b) + b · bWhich is the same as:|a|² + 2(a · b) + |b|²And we know this equals 10 from earlier.Similarly, for |a - b|², it's equal to (a - b) · (a - b). Let's expand that:a · a - a · b - b · a + b · bAgain, since a · b = b · a, this simplifies to:a · a - 2(a · b) + b · bWhich is:|a|² - 2(a · b) + |b|²And this equals 6.So now I have two equations:1. |a|² + 2(a · b) + |b|² = 102. |a|² - 2(a · b) + |b|² = 6Hmm, I have two equations with two unknowns: |a|² and |b|², but I'm actually interested in finding a · b. Maybe I can subtract these equations to eliminate |a|² and |b|².Let's subtract the second equation from the first:(|a|² + 2(a · b) + |b|²) - (|a|² - 2(a · b) + |b|²) = 10 - 6Simplifying the left side:|a|² - |a|² + 2(a · b) - (-2(a · b)) + |b|² - |b|²Which simplifies to:0 + 4(a · b) + 0 = 4(a · b)And the right side is 4.So, 4(a · b) = 4Dividing both sides by 4:a · b = 1So, the dot product of a and b is 1. Looking back at the options, that's option A.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I squared both magnitudes correctly: |a + b|² = 10 and |a - b|² = 6.2. Expanded both using the dot product properties, which seems right.3. Subtracted the second equation from the first to eliminate |a|² and |b|², which gave me 4(a · b) = 4.4. Divided by 4 to get a · b = 1.Everything seems to check out. So, I think the answer is indeed 1, which is option A.**Final Answer**boxed{A}

answer：First, I need to determine the area of the shop by multiplying its length and width.Next, I'll calculate the annual rent by multiplying the monthly rent by 12.Finally, to find the annual rent per square foot, I'll divide the annual rent by the area of the shop.

answer：Okay, so I have this quadratic equation: ( x^2 - 5x + 3 = 0 ). The problem says that the two real roots are ( m ) and ( n ), and I need to find ( m^2 + n^2 ). Hmm, I remember there are some formulas related to the roots of a quadratic equation. Let me think about that.First, I recall that for any quadratic equation of the form ( ax^2 + bx + c = 0 ), the sum of the roots ( m + n ) is equal to ( -frac{b}{a} ), and the product of the roots ( mn ) is equal to ( frac{c}{a} ). So, in this case, ( a = 1 ), ( b = -5 ), and ( c = 3 ).Let me calculate the sum of the roots first. Using the formula ( m + n = -frac{b}{a} ), plugging in the values gives ( m + n = -frac{-5}{1} = 5 ). Okay, so the sum of the roots is 5.Next, the product of the roots ( mn ) is ( frac{c}{a} ), which is ( frac{3}{1} = 3 ). So, the product of the roots is 3.Now, I need to find ( m^2 + n^2 ). I remember there's an identity that relates ( m^2 + n^2 ) to ( (m + n)^2 ) and ( mn ). The identity is ( (m + n)^2 = m^2 + 2mn + n^2 ). If I rearrange this, I can solve for ( m^2 + n^2 ).Subtracting ( 2mn ) from both sides gives ( m^2 + n^2 = (m + n)^2 - 2mn ). That makes sense. So, I can use the values I already have for ( m + n ) and ( mn ) to find ( m^2 + n^2 ).Plugging in the numbers, ( m + n = 5 ) and ( mn = 3 ). So, ( (m + n)^2 = 5^2 = 25 ). Then, ( 2mn = 2 times 3 = 6 ).Subtracting these, ( m^2 + n^2 = 25 - 6 = 19 ). So, the value of ( m^2 + n^2 ) is 19.Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from the quadratic equation, I correctly identified ( a ), ( b ), and ( c ). Then, I applied the sum and product of roots formulas correctly. Calculated ( m + n = 5 ) and ( mn = 3 ). Then, used the identity to find ( m^2 + n^2 ) by squaring the sum and subtracting twice the product. The calculations seem right: 25 minus 6 is indeed 19.I think that's correct. I don't see any errors in my reasoning. So, the final answer should be 19.

answer：Okay, so I have this problem where I need to find the probability that a random variable X is greater than 2 and less than or equal to 4. The probability distribution is given as P(X = k) = 1/(2^k) for k = 1, 2, 3, and so on. Hmm, let me think about how to approach this.First, I know that X is a discrete random variable because it takes on integer values starting from 1. The probability distribution is defined for each integer k, and the probability decreases as k increases since it's 1 over 2 to the power of k. That makes sense because as k gets larger, 2^k grows exponentially, making the probability smaller.Now, I need to find P(2 < X ≤ 4). This means I'm looking for the probability that X is strictly greater than 2 and at most 4. Since X is an integer, the values that satisfy this condition are 3 and 4. So, I need to calculate the probabilities for X = 3 and X = 4 and then add them together.Let me write that down:P(2 < X ≤ 4) = P(X = 3) + P(X = 4)Given the probability distribution, P(X = k) = 1/(2^k), so:P(X = 3) = 1/(2^3) = 1/8P(X = 4) = 1/(2^4) = 1/16Now, I need to add these two probabilities together:P(2 < X ≤ 4) = 1/8 + 1/16To add these fractions, they need a common denominator. The denominators here are 8 and 16. The least common denominator is 16. So, I'll convert 1/8 to sixteenths:1/8 = 2/16Now, adding them together:2/16 + 1/16 = 3/16So, the probability that X is greater than 2 and less than or equal to 4 is 3/16.Wait, let me double-check to make sure I didn't make a mistake. I calculated P(X = 3) as 1/8 and P(X = 4) as 1/16. Adding them gives 3/16. That seems correct.Just to be thorough, I can also think about the total probability distribution. The sum of all probabilities for k = 1 to infinity should be 1. Let's verify that quickly:Sum from k=1 to infinity of 1/(2^k) is a geometric series with first term 1/2 and common ratio 1/2. The sum is (1/2)/(1 - 1/2) = 1. So, that checks out.Therefore, my calculation for P(2 < X ≤ 4) as 3/16 seems accurate.