The locus of the point of intersection of tangents to the ellipse `x^(2)/a^(2) + y^(2)/b^(2) = 1` at the points whose eccentric angles differ by `pi//2`, is

Options

A. `x^(2) + y^(2) = a^(2)`
B. `x^(2) + y^(2) = b^(2)`
C. `x^(2) + y^(2) = a^(2) + b^(2)`
D. `x^(2)/a^(2) + y^(2)/b^(2) = 2`

Answer

D

Explanation

Let `P (a cos theta, b sin theta)` and
`Q (a cos (pi//2 + theta), b sin (pi//2 + theta))` be two points on the ellipse. The equations of tangents to the ellipse at points P and Q are `x/a cos theta + y/b sin theta = 1`
and , `- x/a sin theta + y/b cos theta = 1`
Let P (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) = 2`
`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`.
`ul("ALITER")` The coordinates of the point of intersection of tangents at points having eccentric angles a`alpha and beta` are
`((a cos ((alpha + beta)/(2)))/(cos((alpha + beta)/(2))), ( b sin ((alpha +beta)/(2)))/(cos ((alpha + beta)/(2))))`
Thus, if (h, k) is the point of intersection of tangents at points whose eccentric angles angles differ by `pi/2` i.e. `alpha - beta = pi//2`, then
`h = sqrt(2) a cos ((alpha + beta)/(2)) and k = sqrt2 b sin((alpha + beta)/(2))`
`rArr (h^(2))/(2a^(2)) + (k^(2))/(2b^(2)) = 1`
Hence, the locus of (h, k) is `(x^(2))/(2a^(2)) + (y^(2))/(2b^(2)) = 1`.