Chapter 11.3, Problem 38SR

To determine

To write: An equation for the cross section assuming that the focus is at the origin.

Answer to Problem 38SR

The equation of the parabola is $x=\frac{1}{24}{y}^2-6$ .

Explanation of Solution

Given:

The focus is $6$ inches from the vertex and the parabola opens to the right.

Calculation:

Place the vertex of the parabola at the origin for convenience and use the standard equation that open to the right is given by $x-h=\frac{1}{4a}{\left(y-k\right)}^2$ .

The focus is at $\left(0,0\right)$ so the vertex is at $\left(-6,0\right)$ and also $a=6$ (open to the right).

Substitute $-6$ for $h$ , 0 for $k$ and $6$ for $a$ in the equation $x-h=\frac{1}{4a}{\left(y-k\right)}^2$ .

  $\begin{array}{c}x-\left(-6\right)=\frac{1}{4\left(6\right)}{\left(y-0\right)}^2\\ x+6=\frac{1}{24}{y}^2\\ x=\frac{1}{24}{y}^2-6\end{array}$

Therefore, the equation of the parabola is $x=\frac{1}{24}{y}^2-6$ .