Write the eq. of the circle having its center on the line x=2 and passing through the
point of intersection between the circles of x^2+y^2-3x=0 and x^2+y^2=3y.
You mean "passing through the points of intersection".
Geometrically speaking,
you are looking for a point on the line x=2 that is equidistant from the two points of intersection of the circles, so that both points are a radius away.
Solve for the points of intersection of the circles.
Then use Pythagoras' theorem to find the y co-ordinate of the new circle centre.
Then find that circle's radius.
You can then write the circle equation.