Proving Hyperbola for Quadratic Equation (X^T)AX = k

[blackblanx]

Homework Statement

Let A be a 2x2 symmetric matrix and x be a scalar. Prove that the graph of the quadratic equation (X^T)AX = k is hyperbola if k is non zero and det(A) less than zero

(T stands for transpose

Homework Equations

not were to begin

 

[HallsofIvy]

Well, begin by writing it out!
Let X= <x, y> and
A= \begin{bmatrix} a &amp; b \\ c &amp; d\end{bmatrix}.

Then
X^TAX= \begin{bmatrix}x &amp; y\end{bmatrix}\begin{bmatrix}a &amp; b \\ c &amp; d\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}=ax^2+ cxy+ bxy+ dy^2= ax^2+ (b+ c)xy+ dy^2

Now, under what conditions on a, b, c, d is ax^2+ (b+ c)xy+ dy^2 a hyperbola?