Chapter 10.2, Problem 23AYU

In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand.

Focus at $\left(-2,0\right)$; directrix the line $x\text{ = 2}$

To determine

(a & b)

The equation of the parabola described and two points that define the latus rectum for the following description.

Focus at $\left(-2,\text{ 0}\right)$; directrix the line $x\text{ = 2}$.

Answer to Problem 23AYU

$y^2=-8x$, Latus rectum: $\left(-2,\text{ 4}\right)$ and $\left(-2,-4\right)$.

Explanation of Solution

Given:

Focus at $\left(-2,\text{ 0}\right)$; directrix at the line $x\text{ = 2}$.

Formula used:

Vertex	Focus	Directrix	Equation	Description
$\left(0,0\right)$	$\left(-a,\text{ 0}\right)$	$x=a$	$y^2=-4ax$	Parabola, axis of symmetry is $x\text{-axis}$, opens left

Calculation:

From the given data we see that the focus is at $F\left(-2,\text{ 0}\right)$ and directrix is the line $x=\text{ 2}$. The vertex is $\left(0,0\right)$. Both vertex $V\left(0,0\right)$ and $F\left(-2,\text{ 0}\right)$ lies on the vertical line $y\text{ = 0}$ (The axis of symmetry). The distance $a$ from the vertex $V\left(0,0\right)$ to the focus $F\left(-2,\text{ 0}\right)$ is $a=\text{ 2}$.

Also, because the focus $F\left(-2,\text{ 0}\right)$ lies left of the vertex, the parabola opens left.

$y^2=-4ax$

$a=\text{ 2}$

Therefore the equation is,

$y^2=-4*\text{ 2 }*x$

$y^2=-8x$

To find the points that define the latus rectum, Let $x=-2$ in the above equation.

We get,

$y^2=-8*-2$

$y=\pm 4$

Hence $\left(-2,\text{ 4}\right)$ and $\left(-2,-4\right)$ defines the latus rectum.

To determine

c.

To graph: The parabola for the equation $y^2=-8x$.

Answer to Problem 23AYU

Explanation of Solution

Given:

Focus at $\left(-4,\text{ 0}\right)$; vertex at $(0,0)$.

Calculation:

The graph of the equation of the parabola $y^2=-8x$ is plotted below.

Using the given information D: $x=\text{ 2}$ and $F\left(-2,\text{ 0}\right)$ we have formed the equation and plotted the graph for $y^2=-8x$. The focus $F\left(-2,\text{ 0}\right)$ and the vertex $V\left(0,0\right)$ lie on the horizontal line.

The equation for directrix D: $x=2$ is shown as dotted line. The parabola opens left. The points $\left(-2,-4\right)$ and $\left(-2,\text{ 4}\right)$ define the latus rectum.