Question

At what point on the ellipsoid x2+ y2+2z2 = 1 is the tangent plane parallel to the plane x +2y+z = 1?

Answer 1

First we compute the tangent plane to the ellipsoid at a general point (x0 , y0 , z0 ). The ellipsoid is the level set of f (x, y, z) = x2 + y2 + 2z2 for the value 1. Thus

∇f (x, y, z) = 2xi + 2yj + 4zk

and the tangent plane is given by

because the point (x0 , y0 , z0 ) is assumed to lie on the ellipsoid and thus satisfies x20 + y20 + 2z20 = 1. The question become, for which (x0 , y0 , z0 ) are the planes x0 x + y0 y +2z0 z = 1 and x +2y +z = 1 parallel. Two planes are parallel, if the normal vectors are parallel. So there must exist λ ≠ 0, such that

x0 = λ

y0 = 2λ

2z0 = λ,

and because x20 + y20 + 2z20 = 1, also