Chapter 10.2, Problem 19AYU

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

Answer to Problem 19AYU

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

Explanation of Solution

Given:

Focus at $\left(4,0\right)$; vertex at $\left(0,0\right)$.

Formula used:

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

Calculation:

The vertex $V\left(0,0\right)$ and focus $F\left(4,0\right)$ both lie on the horizontal line $y=0$ (The axis of symmetry). The distance $a$ from the vertex $V\left(0,0\right)$ to the focus $F\left(4,0\right)$ is $a=4$.

Also, because the focus lies to the right of the vertex, the parabola opens to the right.

${\left(y\right)}^2=4a\left(x\right)$; where $a=4$. Therefore the equation is,

${\left(y-0\right)}^2=4*4\left(x-0\right)$

$y^2=16x$

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 $\left(4,8\right)$ 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 19AYU

Explanation of Solution

Given:

Focus at $\left(4,0\right)$; vertex at $\left(0,0\right)$.

Calculation:

Parabola is plotted for the equation $y^2=16x$.

Using the given information $V\left(0,0\right)$ and $F\left(4,0\right)$ we have formed the equation and plotted the graph for $y^2=16x$. The focus $F\left(4,0\right)$ 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 at the right. The points $\left(4,-8\right)$ and $\left(4,8\right)$ define the latus rectum.