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The title is a bit of a mess but here's the deal:

Find a circle that satisfies following conditions:

touches line: $y+2=0$

its centre is on: $x-2y+4=0$

touches internally another circle: $x^2+y^2-2y=0$

I always end up with more variables than equations so I'm kind of stuck knowing there are 2 solutions yet I can't get to them.

Thank you in advance.

Quick note: the previous task was based around the CCP Apollonius's Problem so the solution for this one may or may not have some similarities.

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    There are several ways to attack this problem. If you showed your work, perhaps then someone might point out where you’re going wrong in the approach that you’re taking. – 2017-02-26
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    My first thought was that the tangent lines for the searched circle and the given circle are the same but that doesn't lead anywhere without a specific point. I tried to get to the touching point by expressing it from the central equation (or what's the name for it) knowing that the touching point with the first line can be expressed by given shift of the centre from origin. But that's where I got more variables than equations. – 2017-02-26

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Regarding your first idea, you know that the tangent point lies somewhere on the known circle, so one way to proceed is to take an arbitrary point on this circle and compute the externally tangent circle with center on the given line. The center of this second circle is colinear with the tangent point and center of the known circle, so you can find it by computing the intersection of two lines. Tangency to the line $y+2=0$ means that the unknown circle’s radius must be equal to $|y_c+2|$, where $y_c$ is the $y$-coordinate of that circle’s center. This will give you an equation for the possible tangent points.

You don’t need to compute the point of tangency explicitly, though. The line segment from the center of the unknown circle to this point is a radius of that circle. The two centers and the tangent point are colinear, so its length is equal to the distance between the centers of the circles, plus the radius length of the known circle. So, the center of the unknown circle is a point on the line $x-2y+4=0$ such that this distance is equal to the distance to the line $y+2=0$, i.e., $|y+2|=1+\sqrt{x^2+(y-1)^2}$. This gives you two equations in two unknowns.