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Let E(1) and E(2) be two ellipse whsoe ...

Let `E_(1) and E_(2)` be two ellipse whsoe centers are at the origin. The major axes of `E_(1) and E_(2)` lie along the x-axis , and the y-axis, respectively. Let S be the circle `x^(2)+(y-1)^(2)=2` . The straigth line x+y=3 touches the curves, S, `E_(1) and E_(2)` at P,Q and R, respectively . Suppose that `PQ=PR=(2sqrt(2))/(3)`. If `e_(1) and e_(2)` are the eccentricities of `E_(1) and E_(2)` respectively, thent hecorrect expression (s) is (are)

A

`e_(1)^(2)+e_(2)^(2)=(43)/(40)`

B

`e_(1)e_(2)=(sqrt(7))/(2sqrt(10))`

C

`|e_(1)^(2)-e_(2)^(2)|=(5)/(8)`

D

`e_(1)e_(2)=(sqrt(3))/(7)`

Text Solution

AI Generated Solution

To solve the problem step by step, let's break down the information given and the necessary calculations. ### Step 1: Understand the equations of the ellipses and the circle We have two ellipses \( E_1 \) and \( E_2 \) centered at the origin. The major axis of \( E_1 \) lies along the x-axis, and the major axis of \( E_2 \) lies along the y-axis. The equations can be represented as: - \( E_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (major axis along x-axis) - \( E_2: \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) (major axis along y-axis) The circle \( S \) is given by the equation: ...
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