Chapter 10.2, Problem 22AYU

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

To determine

(a & b)

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

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

Answer to Problem 22AYU

$y^2=-16x$, Latus rectum: $\left(-4,-8\right)$ and $\left(-4,\text{ 8}\right)$.

Explanation of Solution

Given:

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

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:

The vertex $V\left(0,0\right)$ and focus $F\left(-4,\text{ 0}\right)$ both lie on the vertical line $x=0$ (The axis of symmetry). The distance $a$ from the vertex $V\left(0,0\right)$ to the focus $F\left(-4,\text{ 0}\right)$ is $a=\text{ 4}$.

Also, because the focus $F\left(-4,\text{ 0}\right)$ lies left of the vertex, the parabola opens left.

$y^2=-4ax$

$a=\text{ 4}$

Therefore the equation is,

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

$y^2=-16x$

The equation of the parabola is $y^2=-16x$.

To find the points that define the latus rectum, Let $x=-4$ in the above equation.

We get,

$y^2=-16*-4$

$y=\pm 8$

Hence $\left(-4,\text{ 8}\right)$ and $\left(-4,-8\right)$ defines the latus rectum.

To determine

c.

To graph: The parabola for the equation $y^2=-16x$.

Answer to Problem 22AYU

Explanation of Solution

Given:

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

Calculation:

The graph of the equation of the parabola $y^2=-16x$ is plotted below.

Using the given information $V\left(0,0\right)$ and $F\left(-4,\text{ 0}\right)$, we have formed the equation and plotted the graph for $y^2=-16x$. The focus $F\left(-4,\text{ 0}\right)$ and the vertex $V\left(0,0\right)$ lie on the horizontal line.

The equation for directrix D: $x=4$ is shown as dotted line. The parabola opens left. The points $\left(-4,-8\right)$ and $\left(-4,\text{ 8}\right)$ define the latus rectum.