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Question

Let the circle x2+y28x=0 and hyperbola x29y24=1 intersect at the points A and B, where A is in first quadrant. If an ellipse is constructed with vertices at A,B and having eccentricity 13, then the distance between its foci is

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Solution

Given, x2+y28x=0 (1)
x29y24=1 (2)
Solving (1) and (2),
x2+4(x291)8x=09x2+4(x29)72x=013x272x36=0(x6)(13x+6)=0x=6 (x613)
A(6,23) and B(6,23)

Length of major axis of ellipse, 2a=AB=43
e=13
Distance between the foci =2ae=4

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