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

Options

A. `3//13`
B. `4//13`
C. `5//13`
D. none of these

Answer

C

Explanation

We know that the locus of the foot of the perpendicular drawn from foci on any tangent to the ellipse is its auxiliary circle. Therefore ,(1, -4 ) lies on the auxiliary circle of the ellipse.
The foci are S (3, 4) and S' (9, 12). Therefore, co-ordinates of the centre of the ellipse are (6, 8).
Also, Distance between two foci = 10 `rArr 2ae = 10 rArr ae = 5`
The equation of the auxiliary circle is
`(x-6)^(2)+(y-8)^(2)=a^(2)`
It passes through (1, -4). lt brgt ` therefore 25+144=a^(2)rArr a = 13`.
`therefore ae = 5rArr e = 5//13`.