Think you know about the Ellipse? Think again!

Geometry Level 5

Consider an ellipse $E$ with centre $C$ . From a point $K$ , four real, distinct normals are drawn to the ellipse which intersect the major axis of $E$ at $G_1$ , $G_2$ , $G_3$ and $G_4$ . Let $S=\left (\displaystyle\sum_{i=1}^4CG_i \right)\cdot \left (\displaystyle\sum_{i=1}^4\frac{1}{CG_i} \right)$

Let $M=\max(S)$ and $m=\min(S)$ . Find $M-m$

Give your answer to 3 decimal places.

Details and Assumptions :

All lengths are signed.

This problem is part of my set: Geometry

The answer is 0.000.

1 solution

A Former Brilliant Member
Feb 28, 2016

It is worth noting that there is a restriction on the position of $K$ . There will be four, three or two normals that can be drawn through $K$ depending on whether or not the point $K$ lies inside, on, or outside the evolute of the ellipse . Thus $K$ must lie inside the evolute for this question to be possible. You might also want to keep $K$ off the major axis, since then two of the normals are the $x$ -axis itself, and the points $G_1,G_2,G_3,G_4$ are not well-defined. You need to avoid the minor axis as well, since then two of the values $CG_j$ are equal to $0$ , which makes $S$ hard to calculate.

Mark Hennings - 3 years, 2 months ago

Think you know about the Ellipse? Think again!

The answer is 0.000.

1 solution

0 pending reports