Question

A tangent at a point on the circle $x^2+y^2=a^2$ intersects a concentric circle C at two points P and Q. The tangents to the circle X at P and Q meet at a point on the circle $x^2+y^2=b^2$, then the equation of circle is

• A. $x^2+y^2=ab$
• B. $x^2+y^2=(a-b{)}^2$
• C. $x^2+y^2=(a+b{)}^2$
• D. $x^2+y^2=a^2+b^2$

Answer 1

Answer: (A)

$x^2+y^2=ab$

Chord of contact of the point A w.r.t. $x^2+y^2=r^2$ is
$xb\cos \theta =yb\sin \theta =r^2$ .......... (i)
This must be a tangent to the circle $x^2+y^2=a^2$
$\Rightarrow \left[\frac{r^2}{\sqrt{b^2{\cos }^2\theta +b^2{\sin }^2\theta }}\right]=a\Rightarrow r^2=ab$
Hence, equation of circle is $x^2+y^2=ab$.