The locus of the point of intersection of tangents at the end-points of conjugate diameters of the ellipse `x^(2)/a^(2) + y^(2)/b^(2) = 1`, is

Options

A. a circle
B. a parabala
C. an ellipse
D. a hyperbola

Answer

C

Explanation

Let CP and CD be two conjugate semi-diameters of the ellipse `x^(2)/a^(2) + y^(2)/b^(2) = 1`. Then, the eccentric angles of P and D are `theta` and `(pi)/(2) + theta` respectively. So, the coordinates of P and D are `(a cos theta, b sin theta)` and `( - a sin theta, b cos theta)` respectively.
The equations of the tangents at P and D are
`x/a cos theta + y/b sin theta = 1" "...(i)`
and, `- x/a sin theta + y/b cos theta = 1" "...(ii)`
Let (h, k) be the point of intersection of (i) and (ii). Then, `h/a cos theta + k/b sin theta = 1 " and " - h/a sin theta + k/b cos theta = 1`

`rArr (h/a cos theta + k/b sin theta)^(2) + (-h/a sin theta + k/b cos theta)^(2) = 1 + 1`
`rArr h^(2)/a^(2) + k^(2)/b^(2) = 2`
Hence, the locus of (h, k) is `x^(2)/a^(2) + y^(2)/b^(2) = 2` which represents an ellipse.