Chapter 9, Problem 17RE

To determine

To find: Focus, Vertex and Directrix of Parabola.

Answer to Problem 17RE

Vertex of given Parabola = $\left(0,0\right)$

Focus of Parabola = $\left(-9,0\right)$

Directrix of Parabola is $x=9$

Explanation of Solution

Given information:

  $\frac{1}{2}{y}^2+18x=0$

  $\frac{1}{2}{y}^2+18x=0$

  $\begin{array}{l}\frac{1}{2}{y}^2+18x=0\\ \Rightarrow \frac{1}{2}{y}^2=-18x\\ \Rightarrow y^2=-36x\end{array}$

If parabola is in the $y^2=-4ax$

Then

Vertex of Parabola = $\left(0,0\right)$

Focus of Parabola = $\left(-a,0\right)$

Directrix of Parabola is $x=a$

So, on comparing $y^2=-36x$ with standard equation

We get

  $\begin{array}{l}\Rightarrow 4a=36\\ \Rightarrow a=\frac{36}{4}\\ \Rightarrow a=9\end{array}$

Therefore

Vertex of given Parabola = $\left(0,0\right)$

Focus of Parabola = $\left(-9,0\right)$

Directrix of Parabola is $x=9$

Graph of given parabola is

  

Violet line represents the Directrix.