Find out what kind of conic section the following quadratic form represents and transfo

Question

Find out what kind of conic section the following quadratic form represents and transform it to principal axes. Express ${\vec{x}}^{T} = [x_{1} x_{2}]$ in terms of the new coordinate vector ${\vec{y}}^{T} = [y_{1} y_{2}]$
$x_{1}^{2} - 12 x_{1} x_{2} + x_{2}^{2} = 70$

Answer 1

Given quadratic equation is $x_{1}^{2} - 12 x_{1} x_{2} + x_{2}^{2} = 70$
The matrix representation of quadratic form is

Now trace $(A) = 2 and det (A) = - 35$
Characteristic equation of matrix A is $λ^{2} - 2 λ - 35 = 0$
Therefore eigen values of matrix A are $λ = - 5, 7$
Since matrix A has both positive and negative eigen values, therefore matrix A is indefinite.
Given quadratic form is hyperbolic

Find out what kind of conic section the following quadratic form represents and transfo

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