Question

State whether the following statement is true or false.

If the two tangents are mutually perpendicular, the locus of their point of intersection is a circle concentric with the given circles.

• A. True
• B. False

Answer 1

The correct option is A True

Let equation of circle be $x^2+y^2=a^2$.

Then the equation of tangent to the circle $x^2+y^2=a^2$ is

$y=mx+\sqrt{a(1+m^2)}$

$P(h,k)$ lies on tangent then

$(k-mh)=a\sqrt{1+m^2}$

$(k-mh{)}^2=a^2(1+m^2)$

$m^2(h^2-a^2)-2mhk+k^2-a^2=0$

This is the quadratic equation in $m$,let two roots $m_1$ and $m_2$

$m_1{m}_2=-1$ ($\perp$ tangents)

$k^2-a^2{h}^2-a^2=-1$

$k^2-a^2=-h^2+a^2$

$h^2+k^2=2a^2$

Hence locus $P(h,k)$ is $x^2+y^2=2a^2$, which is concentric to given circle.