Question

There exist two points `P` and `Q` on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` such that `PObotOQ`, where `O` is the origin, then the number of points in the `xy`-plane from where pair of perpendicular tangents can be drawn to the hyperbola , is
A. `0`
B. `1`
C. `2`
D. infinite

Answer 1

Let `P(a sectheta_(1),btantheta_(1))` and `Q(a sectheta_(2),btantheta_(2))` be two points on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` such that
`OPbotOQ`
`implies(b)/(a)(tantheta_(1))/(sectheta_(1))xx(b)/(a)(tantheta_(2))/(sectheta_(2))=-1`
`impliessintheta_(1)sintheta_(2)=-(a^(2))/(b^(2))`
`impliesa^(2) lt b^(2)`
The point on `xy`-plane from where perpendicular tangents are drawn to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` lie on the director circle `x^(2)-y^(2)=a^(2)-b^(2)`. For `a^(2) lt b^(2)` the director circle does not exist.So, there is no point in `xy`-plane.