# Geometrical Properties of Ellipses

• Feb 26th 2010, 03:59 AM
Sunyata
Geometrical Properties of Ellipses
Hi.

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.
• Feb 26th 2010, 01:58 PM
running-gag
Hi

The line $y = mx+c$ touches the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ when

$\frac{x^2}{a^2} + \frac{\left(mx+c\right)^2}{b^2} = 1$ has 1 solution

$\left(\frac{1}{a^2}+\frac{m^2}{b^2}\right)x^2 + \frac{2mc}{b^2} x + \frac{c^2}{b^2} - 1 = 0$ has 1 solution

therefore when the discriminant is equal to 0 which gives $c^2 = a^2m^2 + b^2$ after simplification