Chapter 8.2, Problem 51E

Solution

(a)

To explain : The change in the focus, directrix and shape of the graph of the given equation due to the given change in $a$ by graphing the original and revised equations in the same coordinate plane.

The focus changes from $\left(0,\frac{1}{4}\right)$ to $\left(0,1\right)$ and the directrix changes from $y=-\frac{1}{4}$ to $y=-1$ . The revised parabola will be wider than the original one.

Given information :

The equation $x^2=ay$ and $a$ change from $1\text{ to }4$ .

Explanation :

Consider the given equation $x^2=ay$ .

Substitute $a=1$ , so

  $\begin{array}{c}{x}^2=\left(1\right)y\\ =y\end{array}$

The equation is of the form $x^2=4py$ , where $p=\frac{1}{4}$ . Thus the focus is $\left(0,\frac{1}{4}\right)$ and the directrix is $y=-\frac{1}{4}$ .

Now, substitute $a=4$ , so

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

Now, $p=1$ . Thus the focus is $\left(0,1\right)$ and the directrix is $y=-1$ .

The graph of both the equations in the same coordinate plane is shown below:

  

Therefore, the revised parabola is wider than the original one.

(b)

To explain : The change in the focus, directrix and shape of the graph of the given equation due to the given change in $a$ by graphing the original and revised equations in the same coordinate plane.

The focus changes from $\left(\frac{3}{2},0\right)$ to $\left(\frac{1}{8},0\right)$ and the directrix changes from $x=-\frac{3}{2}$ to $x=-\frac{1}{8}$ . The revised parabola will be narrower than the original one.

Given information :

The equation $y^2=ax$ and $a$ change from $6\text{ to }-\frac{1}{2}$ .

Explanation :

Consider the given equation $y^2=ax$ .

Substitute $a=6$ , so

  $\begin{array}{c}{y}^2=\left(6\right)x\\ =6x\end{array}$

The equation is of the form $y^2=4px$ , where $p=\frac{6}{4}\text{ or }\frac{3}{2}$ . Thus the focus is $\left(\frac{3}{2},0\right)$ and the directrix is $x=-\frac{3}{2}$ .

Now, substitute $a=\frac{1}{2}$ , so

  $\begin{array}{c}{y}^2=\left(\frac{1}{2}\right)x\\ =\frac{1}{2}x\end{array}$

Now, $p=\frac{1}{8}$ . Thus the focus is $\left(\frac{1}{8},0\right)$ and the directrix is $x=-\frac{1}{8}$ .

The graph of both the equations in the same coordinate plane is shown below:

  

Therefore, the revised parabola is narrower than the original one.