Question

Let `P(a sectheta, btantheta) and Q(aseccphi , btanphi)` (where `theta+phi=pi/2` be two points on the hyperbola `x^2/a^2-y^2/b^2=1` If `(h, k)` is the point of intersection of the normals at `P and Q` then `k` is equal to (A) `(a^2+b^2)/a` (B) `-((a^2+b^2)/a)` (C) `(a^2+b^2)/b` (D) `-((a^2+b^2)/b)`
A. `(a^(2)+b^(2))/(a)`
B. `-(a^(2)+b^(2))/(a)`
C. `(a^(2)+b^(2))/(b^(2))`
D. `-(a^(2)+b^(2))/(b)`

Answer 1

The coordinates `(h,k)` of the point of intersection of the normals at `P` and `Q` are given by
`h=-((a^(2)+b^(2))/(a))(cos((theta-phi)/(2))costhetacosphi)/(cos((theta+phi)/(2)))`
and, `k=-((a^(2)+b^(2))/(a))tan((theta+phi)/(2))tanthetatanphi`.
We have, `theta+phi=(pi)/(2)`
`k=-((a^(2)+b^(2))/(b))tan.(pi)/(4)tanthetacotthetaimpliesk=-((a^(2)+b^(2))/(b))`