Question

If a variable circle having fixed radius $a$, passes through origin and meets the coordinates axes at point $A$ and $B$ respectively, then the locus of centroid of $△OAB$, where $O$ is the origin, is

Answer 1

Let the variable circle be
$x^2+y^2+2gx+2fy=0$
Putting $x=0$
$y=0,-2f\Rightarrow A=(0,-2f)$
Putting $y=0$
$x=0,-2g\Rightarrow B=(-2g,0)$

Then radius of the circle is
$r=\sqrt{g^2+f^2}\Rightarrow a^2=g^2+f^2$

Let $P(h,k)$ be the centroid of $△OAB$, we get
$h=-\frac{2g}{3},k=-\frac{2f}{3}\because g^2+f^2=a^2\Rightarrow \frac{9h^2}{4}+\frac{9k^2}{4}=a^2$

Hence, the required locus is
$9(x^2+y^2)=4a^2$