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

Question

Chloe Alexander · Accepted Answer

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]}

Theodore Amidon · Answer

undefined To determine the type of conic section represented by the equation $x^2 - 10x - y + 18 = 0$, we can examine the coefficients of $x^2$, $x$, and $y$.

A hyperbola is characterized by equations with terms involving both $x^2$ and $y^2$ with opposite signs.
A parabola typically has either a $x^2$ term or a $y^2$ term, but not both.
A circle has both $x^2$ and $y^2$ terms with the same coefficient.
An ellipse has both $x^2$ and $y^2$ terms with different coefficients, but the signs are the same.

In the given equation, there is only one squared term, which is $x^2$. This indicates that the conic section may be either a parabola or a circle.

To confirm, we can check the coefficients of the linear terms. The coefficient of $x$ is -10, indicating that the axis of symmetry is vertical if it's a parabola. However, if it's a circle, it doesn't provide enough information.

Therefore, without additional information, we cannot definitively determine whether the conic section represented by the given equation is a parabola or a circle.