# How do you classify the conic 4x^2-y^2-8x+4y-9=0?

Aug 8, 2017

It is a hyperbola.

#### Explanation:

$4 {x}^{2} - {y}^{2} - 8 x + 4 y - 9 = 0$

$\Leftrightarrow 4 \left({x}^{2} - 2 x + 1\right) - 4 - \left({y}^{2} + 4 y + 4\right) + 4 - 9 = 0$

or $4 {\left(x - 1\right)}^{2} - {\left(y + 2\right)}^{2} = 9$

or ${\left(x - 1\right)}^{2} / {\left(\frac{3}{2}\right)}^{2} - {\left(y + 2\right)}^{2} / {3}^{2} = 1$

As $4 {x}^{2} - {y}^{2} - 8 x + 4 y - 9 = 0$ can be written as ${\left(x - h\right)}^{2} / {a}^{2} - {\left(y - k\right)}^{2} / {b}^{2} = 1$, it is a hyperbola.

graph{4x^2-y^2-8x+4y-9=0 [-9.46, 10.54, -3.76, 6.24]}

Aug 8, 2017

The General Cartesian form for a conic section is:

$A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$
One may use the value of ${B}^{2} - 4 A C$ to classify it.

#### Explanation:

The given equation

$4 {x}^{2} - {y}^{2} - 8 x + 4 y - 9 = 0$

Fits the General Cartesian form with $A = 4 , B , = 0 , C = - 1 , D = - 8 , E = 4 \mathmr{and} F = - 9$

Refering to the reference section entitled, Discriminant , we compute the value:

${B}^{2} - 4 A C = {0}^{2} - 4 \left(4\right) \left(- 1\right) = 16$

The value is greater than zero. The list of conditions given by reference tells us that the conic section is a hyperbola.