I need to prove that if parabola x2=4py has a chord (not necessarily a focal chord) intersecting it at points A and B, with tangents to the parabola at points A and B that intersect at C, then a line drawn through C and the midpoint of the chord M is parallel to the y-axis. Further, prove that the point D where this line intersects the parabola is the midpoint of line CM.

I have tried to determine the slopes of the tangents in terms of the coordinates at points A and B. I got: Mb=2(Yb/Xb), Ma=2(Ya/Xa) ... substituting these slopes into the equations for the lines & solving simultaneously for Xc gives: Xc=XaXb(Yb-Ya)/(2(XaYb-XbYa)) However, in solving for Xm I got: Xm=(Xa+Xb)/2 ... to show line CM parallel to the y-axix I need to show that Xm=Xc, but my approach didn't work. How can I find the correct method of proof?

Thank you


You are almost there. In your expression for Xc you have Xa, Xb, Ya and Yb. But you know that (Xa,Ya) and (Xb,Yb) lie on the parabola so you can substitute for Ya and Yb and express Xc in terms of Xa and Xb only.

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