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Question

Let S(3,4) and S(9,12) be two foci of an ellipse. If the coordinates of the foot of the perpendicular from focus S to a tangent of the ellipse is (1,4), then the eccentricity of the ellipse is

A
45
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B
57
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C
513
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D
713
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Solution

The correct option is C 513
SS=2ae, where a and e are length of semi-major axis and eccentricity respectively.
(93)2+(124)2=2ae
ae=5
Centre is mid-point of SS
Centre (6,8)
Let the equation of the auxiliary circle be (x6)2+(y8)2=a2
We know that the foot of the perpendicular from the focus on any tangent lies on the auxiliary circle.
So, (1,4) lies on auxiliary circle, i.e., (16)2+(48)2=a2
a=13
Since, ae=5 e=513

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