# How do you identify if the equation y^2-6y=x^2-8 is a parabola, circle, ellipse, or hyperbola and how do you graph it?

Feb 25, 2018

Write the equation in the General Cartesian form for a conic section, $A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$:

Compute the discriminant, $\Delta = {B}^{2} - 4 A C$

#### Explanation:

Write the in the General Cartesian form:

${x}^{2} - {y}^{2} + 6 y - 8 = 0$

Compute the discriminant:

$\Delta = {0}^{2} - 4 \left(1\right) \left(- 1\right)$

$\Delta = 4$

If $\Delta < 0$ then the conic section is an ellipse. The special case where the ellipse is a circle is identified by $A = C$ and $B = 0$.

If $\Delta = 0$ then the conic section is a parabola

If $\Delta > 0$ then the conic section is a hyperbola.

Here is the graph:

$g r a p h \left\{{x}^{2} - {y}^{2} + 6 y - 8 = 0 \left[- 9.87 , 10.13 , - 2.44 , 7.56\right]\right\}$