If `tantheta_1.tantheta_2=a^2/b^2` then the chord Joining two points `theta_1 and theta_2` on the ellipse `x^2/a^2+y^2/b^2=1` will subtend a right angle at (A) focus (B) centre (C) end of the major axis (D) end of the major axis

Options

A. focus
B. centre
C. end of the major axis
D. end of the minor axis

Answer

D

Explanation

Let `P (a cos theta_(1), b sin theta_(1))` and `Q (a cos theta_(2), b sin theta_(2))` be two points on the ellipse.
Then, `m_(1) =` slope of `OP = b/a tan theta_(1)`
and, `m_(2) = ` slope of `OQ = b/a tan theta_(2)`
`therefore m_(1)m_(2) = b^(2)/a^(2) tan theta_(1) tan theta_(2)`
`rArr m_(1)m_(2) = b^(2)/a^(2) xx (-a^(2))/(b^(2)) " " [because tan theta_(1) tan theta_(2) = -a^(2)//b^(2)` (given)]
`rArr m_(1)m_(2) = -1`
`therefore PAQ = pi//2`.
Hence, PQ makes a right angle at the centre of the allipse.