## 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 to the ellipse is (1, –4), then the eccentricity of the ellipse is

### Solution

5/13

Let *e* be the eccentricity of the ellipse. Also, let the lengths of semi-major and minor axes be *a* and *b* respectively.

Then,

*SS*’ = 2*ae*.

Since centre of the ellipse is the mid-point of *SS*’. So, the coordinates of the centre are .

The equation of the auxiliary circle is

We know that the foot of the perpendicular from the focus on any tangent lies on the auxiliary circle. Therefore (1, –4) lies on the auxiliary circle (ii).

Now, *ae* = 5 and *a* = 13

.

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