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 the `Delta PAB` as `P` moves on the circle is (1) A parabola (2) A circle (3) An ellipse (4) A pair of straight lines

Options

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

Answer

B

Explanation

Let Q(h, k) be the centroid of `Delta PAB`. Then,
`h=(x_(1)+r)/(3) and k=(y_(1)+r)/(3)`
`rArr x_(1)=3h-r and y_(1)=3k-r`
Since `P(x_(1), y_(1))` lies on `x^(2)+y^(2)=r^(2)`. Therefore,
`x_(1)^(2)+y_(1)^(2)=r^(2)`
`rArr (3h-r)^(2)+(3k-r)^(2)=r^(2) rArr (h-(r)/(3))^(2)+(k-(r)/(3))^(2)=((r)/(3))^(2)`

Hence, the locus of (h, k) is `(x-(r)/(3))^(2)+(y-(r)/(3))^(2)=((r)/(3))^(2)`
Clearly, it represents a circle.