Circles
Locus of Centroid — Circle through Three Points
nta_pyq_2026_jan
Grade None
Question:
Let a circle of radius 4 pass through the origin O, the points $A(-\sqrt{3}a,0)$ and $B(0,-\sqrt{2}b)$, where $a$ and $b$ are real parameters and $ab\neq0$. Then the locus of the centroid of $\triangle OAB$ is a circle of radius
Step-by-Step Solution
Key Concept: Let centre $C=(h,k)$ with $h^2+k^2=16$. A and B on circle: gives $h=-\sqrt{3}a/2$ and $k=-\sqrt{2}b/2$. Centroid $G=(-\sqrt{3}a/3,-\sqrt{2}b/2\cdot 2/3)=(-\sqrt{3}a/3,-\sqrt{2}b/3)$.
Locus $x^2+y^2=64/9$. Radius $=8/3$.
Correct Answer: 2