Q.Find the equation of the locus of a point, the tangents from which to the circle $x^2+y^2=9$ are at right angles.

The director circle of a given circle is the locus traced by the point of intersection of a pair of mutually perpendicular tangents drawn to it. Starting from the slope-product formula for two tangents from an external point $P(x_1,y_1)$ to $x^2+y^2=a^2$ (Section 6.3.3), imposing perpendicularity means setting that product equal to $-1$:

$\frac{y_1^2-a^2}{x_1^2-a^2}=-1 \implies x_1^2+y_1^2=2a^2$ …