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The chord of contact of a point P w.r.t ...

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 (a) on conjugate hyperbola (b) one of the directrix (c) one of the asymptotes (d) none of these

A

conjugate hyperbola

B

one of the directrix

C

asymptotes

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
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.
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