Prove that the conic $x^2 - 4xy + y^2 -2x -20y -11 = 0$ is a hyperbola and find the centre $(h,k)$

Question

I have to prove that the conic $$x^2 - 4xy + y^2 -2x -20y -11 = 0$$ is a hyperbola and find the centre $(h,k)$.

I proved it is a hyperbola using discriminant $b^2-4ac $ and the answer was greater than zero hence a hyperbola. But I cannot seem to change the equation into the form $(x-h)^2/a^2 - (y-k)^2/b^2=1$ so as to find the centre...

I could finally solve it with everyone's Help

Answer 1

Write first the terms containing $x$ as the beginning of the square of an affine function in $x$ and $y$: $$x^2-4xy-2x= (x-2y-1)^2-(4y+4y^2+1),$$ so that the equation becomes \begin{align} x^2 - 4xy + y^2 -2x -20y -11 &= (x-2y-1)^2-(4y+4y^2+1)+y^2-20y-11 \\ &= (x-2y-1)^2-3(y^2+8y+4)\\ &= (x-2y-1)^2-3\bigl((y+4)^2-16\bigr). \end{align} Can you end the calculations?

Answer 2

Let the linear terms vanish in

$$(x+h)^2-4(x+h)(y+k)+(y+k)^2-2(x+h)-20(y+k)-11.$$

By identification,

$$\begin{cases}2h-4k-2=0,\\2k-4h-20=0.\end{cases}$$

Solve for $(h,k)$ and you have the center.