Question

Let AB be a chord of the circle $x^2+y^2=r^2$ subtending a right angle at the center. Then, the locus of the centroid of triangle PAB as P moves on the circle is

• A. a parabola
• B. a circle
• C. an ellipse
• D. a pair of straight lines

Answer 1

The correct option is B a circle

$x^2+y^2=r^2$ is a circle with centre at (0, 0) and radius r units.


Any arbitrary point P on it is $(rcos\theta ,rsin\theta )$ choosing
A and B as (-r, 0) and (0, -r), respectively.
For the locus of the centroid of $\Delta ABP$,
$(\frac{rcos\theta -r}{3},\frac{rsin\theta -r}{3})\equiv (x,y)\therefore rcos\theta -r=3xandrsin\theta -r=3yorrcos\theta =3x+randrsin\theta =3y+r$
Squaring and adding, we get $(3x+r{)}^2+(3y+r{)}^2=r^2$, which is a circle.