Problems related to ellipsoid and canonical form

Question

Problem:

Ellipsoid/canonical form

Given ƒ(x,y,z) = 5x² + 5y² + 5z² - 2xy - 2xz - 2yz, consider the ellipsoid ƒ(x,y,z) = 1.

e shortest and the largest distance from the origin to the surface of the ellipsoid.

(b) Find the principal axes of the ellipsoid.

(c) Find an orthogonal transformation x = RX such that ƒ(x,y,z) = λ₁X² + λ₂Y² + λ₃Z², thereby reducing the equation of the ellipsoid to canonical form.