Chapter 11.1, Problem 49E

To determine

To find: the equation of the parabola.

Answer to Problem 49E

The equation of parabola is $x^2=-4\sqrt{2}y$ .

Explanation of Solution

The equation of parabola is $x^2=-4\sqrt{2}y$ .

Given information:

From the given figure it can be observed that,

The graph of the parabola opens to the downwards and parabola has vertical axis.

The shaded region area is $8$ .

Calculation:here, the graph of parabola opens to the downwards, so $p<0$ .

Because the parabola has vertical axis,

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

  $x^2-4py$

Area of shaded region area = $8$

The shaded region is rectangle.

So the area of each rectangle made by rectangle = $\text{4}$

So it is shown that the width of the rectangle is two times of length of the rectangle.

So,

  $\begin{array}{l}\text{length}·\text{width=4}\\ \text{length}·\text{2}·\text{length=4 [substitute width by length]}\\ {\left(\text{length}\right)}^2=2\text{ [divide]}\\ \text{length=}\sqrt{2}\text{ [sqaure root]}\end{array}$

The length of the rectangle is equal to the distance of focus from the origin.

So the focus is $\left(0,-\sqrt{2}\right)$

So, $p=\sqrt{2}$

So the equation of parabola is,

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

Hence, the equation of parabola is $x^2=-4\sqrt{2}y$ .