4. Let $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$ be two points in the $(x, y)$-plane such that $y > 0$ and $x_1 < x_2$.
We call a circle $C$ a Yeppeun circle for $P_1$ and $P_2$ if $C$ contains $P_1$ and $P_2$, $C$ is tangent to the $x$-axis at a point $P_3 = (x_3, 0)$, and the $x$-coordinate of the point of tangency lies between the $x$-coordinates of $P_1$ and $P_2$; that is, $x_1 < x_3 < x_2$. For example, the circle $C$ in the following diagram is a Yeppeun circle for the given $P_1$ and $P_2$
Given any two such points $P_1$ and $P_2$ above the $x$-axis, does a Yeppeun circle always exist?
Explain.
