Question

If a circle of constant radius $3$k passes through the origin and meets the axes at A and B, the locus of the centroid of $\Delta$OAB is?

• A. $x^2+y^2=k^2$
• B. $x^2+y^2=2k^2$
• C. $x^2+y^2=3k^2$
• D. $x^2+y^2=4k^2$

Answer 1

The correct option is D $x^2+y^2=4k^2$

Let the center of the circle be $(a,b)$

As the circle passes through the origin,then $a^2+b^2=9k^2$

A coordinates $(x_1,0)$ , B's $(0,y_1)$

$(a-x_1{)}^2+b^2=9k^2$

$-2a{x}_1+x_1^2=0$

$⟹x_1=2a$

Similarily, $y_1=2b$

Centroid of the$△OAB=(\frac{2a}{3},\frac{2b}{3})$

$(\frac{2a}{3}{)}^2+(\frac{2b}{3}{)}^2=\frac{4\times 9k^2}{9}$

$x^2+y^2=4k^2$