Some details

Now give the details of proof: PC/CR = a/c.
PC = sqrt((x-c)^2 + y^2).
Let R be (x_R, 0). Easy to see x_R = xc^2/a^2
CR = c - x_R = c - xc^2/a^2.
PC/CR = (a/c)*K, where K = a*sqrt((x-c)^2 + y^2)/(a^2-xc). Using the ellipse equation, get K = 1.