Question

A circle of constant radius 3 unit passes through origin and meets the axes at A and B. The locus of the centroid of the triangle OAB is

• A. $x^2+y^2=1$
• B. $x^2+y^2=8$
• C. $x^2+y^2=4$
• D. None of these

Answer 1

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

Let the center of the circle be $(h,k)$.

Give the radius is 3 units.

The equation of the circle can be written as $(x-h{)}^2+(y-k{)}^2=9$

The circle passes through origin, so $h^2+k^2=9$

Let the intercepts be $A(a,0),B(0,b)$

substituting the points in the circle equation, $a=2h,b=2k$

Centroid of the triangle $OAB=(a/3,b/3)=(2h/3,2k/3)$

$(2h/3{)}^2+(2k/3{)}^2=4$

So, the locus is $x^2+y^2=4$