<s>[INST] A circle tangent to two circles touching internally and line Find the radius of a circle touching two circle $x^2+y^2+3\sqrt{2}(x+y)=0$ and $x^2+y^2+5\sqrt{2}(x+y)=0$ and also touching the common diameter of the two given circles.
The two circles touch internally and the common diameter is $x-y=0$.
Let centre of required circle be $(h,k)$ 
$$\sqrt{\left(h-\frac{3}{\sqrt{2}}\right)^2+\left(k-\frac{3}{\sqrt{2}}\right)^2}=r+3$$
$$\sqrt{\left(h-\frac{5}{\sqrt{2}}\right)^2+\left(k-\frac{5}{\sqrt{2}}\right)^2}=5-r$$
and 
$$r=\frac{h-k}{\sqrt{2}}$$
assuming $h>k$
Squaring,
$$h^2+k^2+3\sqrt{2}h+3\sqrt{2}k=r^2+6r$$
$$h^2+k^2+5\sqrt{2}h+5\sqrt{2}k=r^2-10r$$
substituting the value of $r$ and subtracting the two equations, 
$$k=\frac{5h}{3}$$
But $h>k$. Where am I making the mistake?
I think there are two such circles which are mirror immages about $y=x$. One of them has $h>k$.
 [/INST] I am posting a diagram based on my understanding of the question.

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