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

1 Answer
Aug 7, 2018

This is a parabola.

Explanation:

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]}