How do you know if #x^2-10x-y+18=0# is a hyperbola, parabola, circle or ellipse?

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Answer 1 · 2018-08-07T22:41:50

This is a parabola.

Given:

#x^2-10x-y+18 = 0#

Note that the only term of degree #> 1# is #x^2#.

Since the multiplier of #y# is non-zero, we can deduce that this equation represents a parabola.

In fact, adding #y# to both sides and transposing, it becomes:

#y = x^2-10x+18#

which clearly expresses #y# as a quadratic function of #x#, and hence a parabola with vertical axis.

We can also complete the square to find:

#y = (x-5)^2-7#

allowing us to identify the vertex #(5, 7)# and axis #x=5#.

graph{(x^2-10x-y+18)(x-5+0.0001y)((x-5)^2+(y+7)^2-0.01) = 0 [-6.455, 13.545, -7.88, 2.12]}