Not sure if someone already asked this, but I am trying to find an equation of a circle. These are the information.
The circle $C$ is tangent to the circle $O$ with an equation $x^{2} + y^{2} = c^{2}$, and to the line $y = y_{1}$, which we name as $\ell_{1}$. Assume that $y_{1} > 0$, and the circle we choose is the "upper" circle.
The solution I made follows.
Suppose that the center of $C$ is at the point $(h,k)$ and a radius $r$. Then, by the properties of circles, the point of tangency of $O$ and $C$, the center of $C$ and the center of $O$ are collinear, and all points lie on the line $y = \frac{k}{h}x$, which we call $\ell_{2}$.
Solving for the intersection of $O$ and $\ell_{2}$, we get the points $\left(\dfrac{ch}{\sqrt{h^{2} + k^{2}}}, \dfrac{c}{k\sqrt{h^{2} + k^{2}}}\right)$ and $\left(-\dfrac{ch}{\sqrt{h^{2} + k^{2}}}, -\dfrac{c}{k\sqrt{h^{2} + k^{2}}}\right)$. However, the condition implies that the point needed is the former point. We name this point as $P_{1}$.
Since $\sqrt{h^{2}+k^{2}}$ is just equal to $c + r$. By substitution, we get that $P_{1}$ is $\left(\dfrac{ch}{c + r}, \dfrac{c}{k(c + r)}\right)$. Also, since $C$ is tangent to $\ell_{1}$, then it follows that the radius of $C$ is $k - y_{1}$. Thus, $P_{1} = \left(\dfrac{ch}{c + k - y_{1}}, \dfrac{c}{k(c + k - y_{1})}\right)$. This is the point where I was confused. How do I find the values of $h$, $k$, and $r$, in terms of $c$ and $y_{1}$? Or the given information is not enough?
There can be infinitely many circles. Play with the following graph to visualize it. You will get an answer.
Cicrles tangent to another circle and a line
Note that the figure on which the centre of the circle moves is a parabola.