give a picture of six normal are drawn from the external point to the ellipsoid.

Question

I can't imagine this ,somebody help me.If you unable to give this picture then please you give a picture such that three normals are drawn from a external point to the parabola.

Answer 1

Here's a diagram in the case of a parabola. Fixed a point $P=(a,b)$ Let $Q=(x,x^2)$ be a point on the parabola. Line $PQ$ has a slope $(b-x^2)/(a-x)$ and it is perpendicular to the tangent at $Q$ if: $$ {b-x^2\over a-x}\cdot 2x=-1. $$

This is a cubic equation: in order to have three real solutions choose $a$ and $b$ so that its discriminant is positive. In the diagram I chose $P=(2,3)$.