Chapter 8.1, Problem 15E

Solution

To calculate : The equation in the standard form for the parabola that satisfies conditions focus $\left(0,5\right)$ and directrix $y=-5$ .

The equation in the standard form for the parabola that satisfies conditions focus $\left(0,5\right)$ and directrix $y=-5$ is $x^2=20y$ .

Given information :

The parabola with focus $\left(0,5\right)$ and directrix $y=-5$ .

Formula used :

The equation for parabola for which focus $\left(0,p\right)$ is above the vertex is $x^2=4py$ .

Calculation :

Consider the parabola with focus $\left(0,5\right)$ and directrix $y=-5$ ,

It can be seen that the focus and directrix are at equidistant from the vertex at $\left(0,0\right)$ .

As the focus is above the vertex, the parabola opens upwards.

The equation of parabola is $x^2=4py$ .

As the focus is at $\left(0,5\right)$ , then $p=5$ . Substitute 5 for $p$ into the equation $x^2=4py$ .

  $\begin{array}{c}{x}^2=4\left(5\right)y\\ =20y\end{array}$

Thus, the equation for parabola for which focus $\left(0,p\right)$ is above the vertex is $x^2=20y$ .