Question

Two perpendicular tangents to the circle $x^2+y^2=a^2$ meet at $P$. Then the locus of $P$ has the equation

• A. $x^2+y^2=2a^2$
• B. $x^2+y^2=3a^2$
• C. $x^2+y^2=4a^2$
• D. none of these

Answer 1

The correct option is A $x^2+y^2=2a^2$

Consider the point $P(h,k)$.

Since the tangents are mutually perpendicular $\angle P={90}^0$

Therefore $\angle \frac{P}{2}={45}^0$.

Let the point of contact of the tangent and the circle be A and B.

And let the center of the circle be C.

Then in $\Delta PAC$,

$PA=CA$ since the triangles are isosceles right angled triangle.

Therefore

$PC=\sqrt{2}a$

Hence the center is the origin, hence

$P{C}^2=2a^2$

Or

$h^2+k^2=2a^2$
Replacing $(h,k)$ by $(x,y)$, we get

$x^2+y^2=2a^2$