Circle 1 Question 8
8. Let $A B$ be a chord of the circle $x^{2}+y^{2}=r^{2}$ subtending a right angle at the centre. Then, the locus of the centroid of the $\triangle P A B$ as $P$ moves on the circle, is
$(2001,1$ M)
(a) a parabola
(b) a circle
(c) an ellipse
(d) a pair of straight lines
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Answer:
Correct Answer: 8. (a)
Solution:
- Choosing $O A$ as $X$-axis, $A=(r, 0), B=(0, r)$ and any point $P$ on the circle is $(r \cos \theta, r \sin \theta)$. If $(x, y)$ is the centroid of $\triangle P A B$, then
$$ \begin{array}{ll} \text { and } & 3 y=r \sin \theta+0+r \\ \therefore & (3 x-r)^{2}+(3 y-r)^{2}=r^{2} \end{array} $$
Hence, locus of $P$ is a circle.
