Chapter 11.1, Problem 40E

To determine

To find: the equation of the parabola.

Answer to Problem 40E

The equation of parabola is $x^2=-8y$ .

Explanation of Solution

The equation of parabola is $x^2=-8y$ .

Given information:

Vertex $\left(0,0\right)$

Focus on the negativey-axis

Focal diameter = $8$

Calculation:because it is given that focus is on the negativey-axis, so the parabola should be with vertical axis and $p<0$ .

So the equation of parabola should be in the following form,

  $x^2-4py$

Focal diameter of parabola = $\left|4p\right|$

Since focal diameter of parabola = $8$

So,

  $\begin{array}{l}\left|4p\right|=8\\ -\left(4p\right)=8\text{ [remove the mod, because p<0 so it remove with negative sign]}\\ p=-2\text{ [divide by -4]}\end{array}$

So the equation of parabola is,

  $\begin{array}{l}{x}^2-4py\\ x^2-4\left(-2\right)y\text{ [substitute p by -2]}\\ x^2+8y\text{ [multiply]}\\ x^2+8y=0\text{ [simplify]}\\ x^2=-8y\end{array}$

Hence, the equation of parabola is $x^2=-8y$ .