Given two equations of a circle with known radii and $R_2 > R_1$, the circle with the smaller radius is internally tangent with the larger circle. Find the equation of the smaller circle such that is passes through a point, $P_1$, that is inside the larger circle. Only the center of the larger circle is known. Assume that $P_1$ makes such an equation possible.
So, I generally have an good enough approach but may be too explicit to cover the problem.
- Consider the equation: $(x-h_2)^2 + (y-k_2)^2 = (R_2 - R_1)^2$. the center of the circle should lie in this equation.
- Express the previous explicitly as a center to form the general equation of the smaller circle.
- Consider the equation of the larger circle: $(x-h_2)^2 + (y-k_2)^2 = R_2^2$. Find explicit form of the tangent point using the previous 2 equations.
- Find the center of the smaller circle such that distance between the tangent point and $P_1$ is equal.
While I found the general process to be short, it becomes quite messy when the center of the larger circle is not in the origin and when expressing points explicitly.