The chord of contact of a point `P` w.r.t a hyperbola and its auxiliary circle are at right angle. Then the point `P` lies on conjugate hyperbola one of the directrix one of the asymptotes (d) none of these

Options

A. conjugate hyperbola
B. one of the directrix
C. asymptotes
D. none of these

Answer

C

Explanation

Let `P(h,k)` be any point. The chord of contact of P w.r.t. the hyperbola is
`(hx)/(a^(2))-(ky)/(b^(2))=1" (1)"`
The chord of contact of P w.r.t. the auxxiliary circle is
`hx+ky=a^(2)" (2)"`
Now, `(h)/(a^(2))xx(b^(2))/(k)xx(-(h)/(k))=-1`
`"or "(h^(2))/(a^(2))-(k^(2))/(b^(2))=0`
Therefore, P lies on one of the asymptotes.