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If e(1) and e(2) are eccentricities of t...

If `e_(1) and e_(2)` are eccentricities of the hyperbolas `xy=c^(2) and x^2-y^(2)=a^(2)` then `e_(1)^(2)+e_(2)^(2)=`

A

4

B

1

C

`e_(1)^(2)-e_(2)^(2)`

D

`2e_(1)^(2) e_(2)^(2)`

Text Solution

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The correct Answer is:
To solve the problem, we need to find the eccentricities \( e_1 \) and \( e_2 \) of the given hyperbolas and then compute \( e_1^2 + e_2^2 \). ### Step 1: Identify the first hyperbola The first hyperbola is given by the equation: \[ xy = c^2 \] This is a rectangular hyperbola. ### Step 2: Find the eccentricity \( e_1 \) For a rectangular hyperbola, the eccentricity \( e_1 \) is given by: \[ e_1 = \sqrt{2} \] ### Step 3: Identify the second hyperbola The second hyperbola is given by the equation: \[ x^2 - y^2 = a^2 \] ### Step 4: Find the eccentricity \( e_2 \) To find the eccentricity of this hyperbola, we can rewrite it in the standard form: \[ \frac{x^2}{a^2} - \frac{y^2}{a^2} = 1 \] Here, we can see that \( a^2 = a^2 \) and \( b^2 = a^2 \) (since it is a hyperbola with equal semi-major and semi-minor axes). The eccentricity \( e_2 \) for a hyperbola is given by: \[ e_2 = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + 1} = \sqrt{2} \] ### Step 5: Calculate \( e_1^2 + e_2^2 \) Now we can find \( e_1^2 + e_2^2 \): \[ e_1^2 = (\sqrt{2})^2 = 2 \] \[ e_2^2 = (\sqrt{2})^2 = 2 \] Thus, \[ e_1^2 + e_2^2 = 2 + 2 = 4 \] ### Final Answer The required value of \( e_1^2 + e_2^2 \) is: \[ \boxed{4} \]
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