Chapter 13.5, Problem 52STP

To determine

To find: Whether the graph of each equation is parabola, circle, ellipse, or hyperbola.

Answer to Problem 52STP

Graph of given equation is Ellipse

Explanation of Solution

Given information:

  $y^2+18y-2x=-84$

Concept:

Equation in the form

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Then we need to calculate discriminant

  $\Delta =B^2-4AC$

If $\Delta =0$ and $A=0$ or $C=0$

Then the figure is Parabola

If

  $\Delta >0$

Then the figure is Hyperbola

If $\Delta <0$ and $A=C$

Then the figure is Circle.

If $\Delta <0$ and $A\ne C$

Then the figure is Ellipse.

Given

  $y^2+18y-2x=-84$

On simplifying

We get

  $y^2-2x+18y+84=0$

On comparing with standard equation

We get

  $\begin{array}{l}A=0\\ B=0\\ C=1\\ D=-2\\ E=18\\ F=84\end{array}$

Substituting values in discriminant formula

We get

  $\Rightarrow \Delta =0^2-4\left(0\right)\left(1\right)=0$

Here

$\Delta =0$ and $A=0$

So, given equation is Ellipse.