The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length $3a$ is

As the centre of the circumcircle of an equilateral triangle is its centroid and the distance of the centroid from the vertex of the triangle is 2/3 of its median through that vertex. So the distance of the centre from a vertex of the triangle is $(2/3)\times 3a=2a$ and hence the equation of the  circumcircle is $x^2+y^2=4a^2$.