If $P$ is a point of a ellipse, show that the circle with diameter $PF$ ($F$ is one of the foci) is tangent to the circle with diameter $2a$ ( $a$ is the major semi-axis of the ellipse) and center coinciding with the center of the ellipse.
I tried to do it using analytical geometry. I used with out loss of generality a ellipse with semi-axi a parallel to the x-axis and center in origin. Then you have $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ you can find the focus , and the vertices. With that you can write the equation of the circle with center in the origin and radius a. You then use a generic point with coordinates $xp$ and $yp$ that lies in the ellipse. Finally you find the equation of the circle with diameter $PF$ and then show that the distance from the origin (center of big circle) is equal to the difference of the radius of these two circles. I could do that way, but it's too long. I would like to see if there is a shorter solution
