Question

If two perpendicular tangents can be drawn from the origin to the circle $x^2-6x+y^2-2py+17=0$, then the value of $p^2$ is_______.

Answer 1

Answer: 25

The equation of given circle is $x^2+y^2-6x-2py+17=0$
or $(x-3{)}^2+(y-p{)}^2=\left(p^2-8\right)$ ....(i)
Also $(0,0)$ lies outside the circle.
Equation of director circle of $S=0$ will be
$(x-3{)}^2+(y-p{)}^2=2\left(p^2-8\right)$ ....(ii)
Tangents drawn from $(0,0)$ to circle (i) are perpendicular to each other.
$\therefore (0,0)$ must lie on director circle.
$\therefore (0-3{)}^2+(0-p{)}^2=2\left(p^2-8\right)$
$\Rightarrow p^2=25$
$\Rightarrow p=\pm 5$