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Q.
Let $A B$ be a chord of the circle $x^2+y^2=r^2$ subtending a right angle at the centre. Then, locus of the centroid of the triangle $PAB$ as $P$ moves on the circles is
Conic Sections
Solution:
Let $A(r, 0)$ and $B(0, r)$ be end points of chord $A B$ and moving point $P(r \cos \theta, r \sin \theta)$ on the circle.
Let centroid is $(x, y)$
$\Rightarrow x=\frac{r+r \cos \theta+0}{3} $
$\Rightarrow (3 x-r)^2=r^2 \cos ^2 \theta ........$(1)
$\Rightarrow y=\frac{0+r \sin \theta+r}{3}$
$\Rightarrow (3 y-r)^2=r^2 \sin ^2 \theta ........$(2)
Equation (1) $+(2)$
$\Rightarrow (3 x-r)^2+(3 y-r)^2=r^2 $ which is a circle
