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.

(In reply to A note about the problem. by Jer)

With your parabola y = x^2, what about points
 
   E = (-sqrt(b),b)
   F = (sqrt(k*b),k*b)
   G = (k*sqrt(b),k^2*b)
 
with k,b > 0. The y coordinates form the
geometric sequence.
 
The intersection of the tangent lines to
points E and G would have an y coordinate
that is less than zero.

Posted by Bractals on 2013-02-03 15:50:03