Question

The normal at a point `P` of the ellipsoid `x^2/a^2+y^2/b^2+z^2/c^2=1` meets the coordinate planes in `A,B,C`. Show that `AP:BP:CP::a^2:b^2:c^2`?

The normal at a point `P` of the ellipsoid `x^2/a^2+y^2/b^2+z^2/c^2=1` meets the coordinate planes in `A,B,C`. Show that `AP:BP:CP::a^2:b^2:c^2`

Answer 1

See below.

Explanation:

The normal at a point `P` to the ellipsoid `x^2/a^2+y^2/b^2+z^2/c^2=1` meets the coordinate planes in `A,B,C`. Show that `AP:BP:CP::a^2:b^2:c^2`

Given `E->x^2/a^2+y^2/b^2+z^2/c^2-1=0`

and a point `p_0=(x_0,y_0,z_0) in E`

the normal to `E` at `p_0` is given by

`L->p=p_0+lambda vec n_0`

where `lambda in RR` and `vec n_0 = 1/2 (grad E)_(p_0) = (x_0/a^2,y_0/b^2,z_0/c^2)`

The intersections with the planes are given by

`{(0=x_0+lambda_x x_0/a^2),(0=y_0+lambda_y y_0/b^2),(0=z_0+lambda_z z_0/c^2):}`

for values of `lambda = {-a^2,-b^2,-c^2}` The intersection points then are:

`A = (0, y_0-a^2y_0/b^2,z_0-a^2z_0/c^2)`
`B=(x_0-b^2x_0/a^2,0,z_0-b^2 z_0/c^2)`
`C=(x_0 -c^2x_0/a^2,y_0-c^2y_0/b^2,0)`

and

`(PA)/(PB)=a^2/b^2`
`(PA)/(PC)=a^2/c^2`
`(PB)/(PC)=b^2/c^2`