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?

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Answer 1 · 2018-02-25T12:46:20

Write the equation in the General Cartesian form for a conic section, $Ax^2+Bxy+Cy^2+Dx+Ey+F =0$ :

Compute the discriminant, $Delta = B^2-4AC$

Write the in the General Cartesian form:

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

Compute the discriminant:

$Delta = 0^2-4(1)(-1)$

$Delta = 4$

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:

$graph{x^2-y^2+6y-8 = 0 [-9.87, 10.13, -2.44, 7.56]}$

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