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]} #