Chapter 11.1, Problem 46E

To determine

To find: the equation of the parabola.

Answer to Problem 46E

The equation of parabola is $x^2=10y$ .

Explanation of Solution

The equation of parabola is $x^2=10y$ .

Given information:

From the given figure it can be observed that

Focal diameter = $5+5$

The graph of the parabola opens upwards.

The focus of positive is on the positivey-axis.

Calculation:because it is given that focus is on the positivey-axis, so the parabola should be with vertical axis and because the graph of parabola opens upward so $p>0$ .

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 = $5+5$

So,

  $\begin{array}{l}\left|4p\right|=5+5\\ \left|4p\right|=10\text{ [add]}\\ \left(4p\right)=10\text{ [remove the mod, because p>0 so it remove with positive sign]}\\ p=\frac{10}{4}\text{ [divide by 4]}\\ \text{p=}\frac{5}{2}\text{ [simplify]}\end{array}$

So the equation of parabola is,

  $\begin{array}{l}{x}^2-4py\\ x^2-4\left(\frac{5}{2}\right)\text{y [substitute p by }\frac{5}{2}\text{]}\\ x^2-10y\text{ [multiply and divide]}\\ x^2-10y=0\text{ [simplify]}\\ x^2=10y\end{array}$

Hence, the equation of parabola is $x^2=10y$ .