Geometrical Properties of Ellipses
Could someone please help me with these ellipse problems?
1. If M and M' are the points where the tangent at P on the ellipse meets the tangents at the ends of the major axis, then MM' subtends a right angle at either focus. Show that the line y = mx + c touches the ellipse x^2/a^2 + y^2/b^2 = 1 when c = +/- sqrt[a^2m^2+b^2]
2. 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.
3. Show that the semi-latus rectum of the ellipse (acosx, bsiny) is of length b^2/a.
4. The extremities of any diameter of an ellipse are L, L' and P is any point on the ellipse. Show that the product of the gradients of the chords LP and L'P are constant.