locus of centre of a moving circle which orthogonal to another circle

Question

Suppose that a moving circle is orthogonal to the circle $x^2+y^2+2gx+2fy+c=0$ and tangent to the x-axis, find the equation of the locus of the centre of the moving circle.

How to start? Do I need to use the equation $2gg'+2ff'=c+c'$?

Answer 1

Let equation of required circle be $$x^2 +y^2+2g'x+2f'y+c'=0.$$ Let center of required circle be $$(-g',-f').$$ As it is tangent to x-axis it's radius will be equal to modulus of its ordinate. Thus we can say that $$(g')^2 +(f')^2 - c' = (f')^2$$ Thus Hence$$ 2g*g'+2f*f'=c+g'^2$$ Hence Now replace $-g'$ and $-f' $with $x $ and $y$.

Therefore $$x^2+2gx+2fy+c=0$$