<s>[INST] Find the equation of a circle tangent to a circle and x-axis, with center on a certain line. Find the equation of the circle, which is tangent to the circle $x^2 + y^2=4$ and $x-axis$, with center on the line $x=4$.
So far I have:$$(x-h)^2 + (y-k)^2=r^2$$
Since the center is on the line $x=4$, then the center is $(4;k)$.
$\therefore ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(x-4)^2 + (y-k)^2=r^2$ 
And because the circle touches the $x-axis$, $r=k$.
$\therefore~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(x-4)^2 + (y-k)^2=k^2$
Now, I am struggling to use $x^2 + y^2=4$ to find $k$.
Please help.
 [/INST] let $$B(xB,yB)$$ the common Point of the two circles then we have
$$xB^2+yB^2=4$$
$$(xB-4)^2+(yB-yM)^2=yM^2$$
solving this System we get
$$\frac{10-4xB)^2}{yM}=yB$$
from here we get the equation
$$xB^2+\left(\frac{10-4xB}{y_M}\right)^2=4$$
the dicriminant of this equation must be Zero, thus we have
$$yM^2(yM^2-9)=0$$ and we get $$yM=\pm 3$$
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