Question

Let $AB$ 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 $PAB$ as $P$ moves on the circle is

• A. A parabola
• B. A circle
• C. An ellipse
• D. A pair of straight line

Answer 1

The correct option is B A circle

Let $x+y=r$ is a chord that subtends a right angle at origin.

Let $P(h,k)$ be on circle

Then, $h^2+k^2=r^2$ ...... $\left(i\right)$

Then centroid of $PAB$

$x=\frac{r+h}{3}$ and $y=\frac{r+k}{3}$

So, $h=3x-r$ and $k=3y-r$

$\therefore (3x-r{)}^2+(3y-r{)}^2=r^2$

$9(x^2+y^2)-6xr-6yr+r^2=0$

Thus, the locus of the centroid of $PAB$ is a circle.