Chapter 10.2, Problem 20AYU

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 $\left(0,\text{ 2}\right)$; 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 $\left(0,2\right)$; vertex at $\left(0,0\right)$.

Answer to Problem 20AYU

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

Explanation of Solution

Given:

Focus at $\left(0,2\right)$; vertex at $\left(0,0\right)$.

Formula used:

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

Calculation:

The vertex $V\left(0,0\right)$ and focus $F\left(0,2\right)$ 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 $F\left(0,2\right)$ is $a=2$.

Also, because the focus $F\left(0,2\right)$ lies above the vertex, the parabola opens up.

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

$x^2=4*2\left(y\right)$

$x^2=8y$

The equation of the parabola is $x^2=8y$.

To find the points that define the latus rectum, Let us substitute $y=2$ in the above equation.

$x^2=8*2$

$x=\pm 4$

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

To determine

c.

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

Answer to Problem 20AYU

Explanation of Solution

Given:

Focus at $\left(0,2\right)$; vertex at $\left(0,0\right)$.

Calculation:

The graph of $x^2=8y$ is plotted below.

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

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