The locus of point of intersection of tangents to an ellipse `x^2/a^2+y^2/b^2=1` at two points the sum of whose eccentric angles is constant is

Options

A. parabola
B. circle
C. ellipse
D. straight line

Answer

D

Explanation

The equation of tangents at two points having eccentric angles `theta_(1)` and `theta_(2)` are
`x/a cos theta_(1) + y/b sin theta_(2) = 1" "…(i)`
and, `x/a cos theta_(2) + y/b sin theta_(2) = 1`
The point of intersection of (i) and (ii) is
`((a cos ((theta_(1) + theta_(2))/(2)))/(cos ((theta_(1) - theta_(2))/(2))), (b sin ((theta_(1) + theta_(2))/(2)))/(cos ((theta_(1) - theta_(2))/(2))))`
It is given that `theta_(1) + theta_(2) = k =` constant. Therefore, if `(x_(1), y_(1))` is the points of intersection of (i) and (ii), then
`x_(1) = (a cos k)/(cos ((theta_(1) - theta_(2))/(2)))` and `y_(1) = (b sin k)/(cos ((theta_(1) - theta_(2))/(2)))`
`rArr x_(1)/y_(1) = q/b cot k rArr y_(1) = (b/a cot k)x_(1)`
`rArr (x_(1), y_(1))` lies on the straight line `y = (b/a cot k)x.`