Thank you for reporting, we will resolve it shortly
Q.
If two perpendicular tangents can be drawn from the origin to the circle $x^{2}-6 x+y^{2}-2 p y+17=0$, then the value of $p^{2}$ is_______.
Conic Sections
Solution:
The equation of given circle is $x^{2}+y^{2}-6 x-2 p y+17=0$
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 $
$\Rightarrow p=\pm 5$