Question

A variable circle having fixed radius ${}^{\prime }{a}^{\prime }$ passes through the origin and meets the coordinates axes in point $A$ and $B$. Locus of centroid of $△OAB$, $O$ being the origin, is:

• A. $9(x^2+y^2)=4a^2$
• B. $9{(x}^2+y^2)=a^2$
• C. $9(x^2+y^2)=2a^2$
• D. $9{(x}^2+y^2)=8a^2$

Answer 1

Answer: (A)

$9(x^2+y^2)=4a^2$

Let centroid of $△ABC$(isosceles triangle) is h.k as,

$h=\frac{\sqrt{2}a}{3}⟶\left(1\right)$

$k=\frac{\sqrt{2}a}{3}⟶\left(2\right)$

Squaring and adding both (1) and (2)

We get,

$9h^2+{9k}^2={2a}^2+{2a}^2$

$9{(h}^2+k^2)={4a}^2⟶(3)$

replace (h,k)$⟶$(x,y) in equation (3)

$9{(h}^2+k^2)={4a}^2$