Chapter 9.1, Problem 55E

To determine

To find: The standard form of the equation of the parabola with vertex at origin and directrix $y=4$ .

Answer to Problem 55E

The equation of the parabola with vertex at origin and directrix $y=4$ is $x^2=-16y$ .

Explanation of Solution

Given information:

The vertex of the parabola is at origin and directrix $y=4$ .

Calculation:

The directrix parabola is parallel to $x$ -axis. So the parabola is vertical axis parabola with vertex at origin and directrix $y=4$ .

The standard form of general equation of a vertical axis parabola is ${\left(x-h\right)}^2=4p\left(y-k\right)$ .

The general equation of directrix for vertical axis parabola is $y=k-p$ .

Substitute $0$ for $k$ in the general equation of directrix.

  $\begin{array}{l}y=k-p\\ y=0-p\\ y=-p\end{array}$

As given the directrix of parabola is $y=4$ . This gives the value of $p$ as $-4$ .

Substitute $0$ , $0$ for $h$ , $k$ and $-4$ for $p$ in the general equation.

  $\begin{array}{c}{\left(x-h\right)}^2=4p\left(y-k\right)\\ {\left(x-0\right)}^2=4\left(-4\right)\left(y-0\right)\\ x^2=-16y\end{array}$

Therefore, the equation of the parabola with vertex at origin and directrix $y=4$ is $x^2=-16y$ .