Question

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

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

Answer 1

The correct option is C a circle

AB = a is any chord

C(h,k) is the midpoint of AB

$⟹AC=BC=\frac{a}{2}$

Given $\angle AOB=90$

$⟹A{B}^2=O{A}^2+O{B}^2$

$⟹a^2=r^2+r^2=2r^2$

In $△OBC$

$OC\perp BC$

$⟹O{C}^2=O{B}^2-B{C}^2$

$(h-0{)}^2+(k-0{)}^2=r^2–(\frac{a}{2}{)}^2$

$h^2+k^2=r-\frac{r^2}{2}=\frac{r^2}{2}$

Locus of above equation is

$x^2+y^2=(\frac{r}{\sqrt{2}}{)}^2$ which is a circle