Hi
How exactly would I do this?
The normal to the ellipse x^2/a^2 + y^2/b^2 = 1 at P(x1, y1) meets the x-axis in N and the Y-axis in G. Prove PN/NG = (1-e^2)/e^2
See this thread for some general comments about ellipses that should help with this problem. Remember that the eccentricity is given by .
I'm not entirely surprised. This is a longer and messier calculation than I would have guessed. Here is an outline of how to do it.
Take P to be the point . Then (see the link in my other comment above) the normal at P has equation .
Put y = 0 in that equation to see that N is the point . Put x = 0 to see that G is the point .
Then use the usual distance formula to see that .
Similarly .
Therefore , and so .