Characterize the ellipsoid described by ...?

center is (2,5,3)

semi-axes are 3, √6, √3

and axes have directions, respectively

(1,2,2),(-2,-1,2),(2,-2,1)

I procedeed in this way

first I found the center, via translation

x = X + n; y = Y + n; z = Z + p

5(X + m)^2 + 6(Y + n)^2 + 7(Z + p)^2 + 4(X + m)(Y + n) + 4(Y + n)(Z + p)

- 40(X + m) - 80(Y + n) - 62(Z + p) + 37 = 0

5X^2 + 4XY + 2X(5m + 2n - 20) + 6Y^2 + 4YZ + 4Y(m + 3n + p - 20) + 7Z^2

+ Z(4n + 14p - 62) + 5m^2 + m(4n - 40) + 6n^2 + n(4p - 80) + 7p^2 - 62p + 37 = 0

setting = 0 all the coefficient of first degree terms I got the system

5m + 2n - 20 = 0

m + 3n + p - 20 = 0

4n + 14p - 62 = 0

m = 2, n = 5, p = 3

so

C(2, 5, 3) is the center

then I looked for the 3x3 matrix A such that

([X;Y;Z] - [2;5;3])' A ([X;Y;Z] - [2;5;3]) = 1

see

http://en.wikipedia.org/wiki/Ellipsoid#Generalised...

and I found

A =

5 2 0

2 6 2

0 2 7

eigenvalues are 1/a^2, 1/b^2, 1/c^2

eigenvectors are the directions of the axis of the ellipsoid

therefore, in a suitable coordinate system (ξ, υ, ζ)

the canonical equation of the ellipsoid is

ξ²/9 + υ²/6 + ζ²/3 = 1