Chapter 8.1, Problem 12E

Solution

To find: An equation in standard form for the parabola that satisfies the given conditions.

The required equation is $x^2=8y$

Given: Vertex $=\left(0,0\right)$ , focus $=\left(0,2\right)$

Concept used:

For Vertex: $\left(h,k\right)$ and Focus: $\left(h,k+p\right)$

The standard equation: ${\left(x-h\right)}^2=4p\left(y-k\right)$

Calculation:

The axis of the parabola is the line passing through vertex $\left(0,0\right)$ and focus $\left(0,2\right)$ .

And its equation is $x=0$ .

The directed distance from the vertex to focus is $p=2-0$ .That is $p=2$ .

Thus, $4p=8$

So, the required equation will be

  $\begin{array}{c}{\left(x-0\right)}^2=8\left(y-0\right)\\ x^2=8y\end{array}$