Three points E, F and G are taken on the parabola y²= 4ax, so that their ordinates (in order) are in geometric sequence. The ordinates of each of the three points have the same sign.

Do the tangents at E and G intersect on the line through F perpendicular to the axis of the parabola?

If this is always so, prove it. Otherwise, provide a counter example.

When this problem was being vetted someone suggested adding the phrase:

"The ordinates of each of the three points have the same sign."

This sentence isn't really needed.  If the ratio of the geometric sequence (k in my proof) is negative all this does is put point F on the opposite side of the y-axis from E and G, but the y-coordinate is the same.  Note in the proof k only ever appears as kČor k^4.

Posted by Jer on 2013-02-03 01:17:48