Chapter 10.2, Problem 21AYU

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 $\left(0,-\text{3}\right)$; 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 $\left(0,-3\right)$; vertex at $\left(0,0\right)$.

Answer to Problem 21AYU

$x^2=-12y$, Latus rectum: $\left(6,-3\right)$ and $\left(-6,-3\right)$

Explanation of Solution

Given:

Focus at $\left(0,-3\right)$; 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 down

Calculation:

The vertex $V\left(0,0\right)$ and focus $F\left(0,-3\right)$ 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 $F\left(0,-3\right)$ is $a=3$.

Also, because the focus $F\left(0,-3\right)$ lies below the vertex, the parabola opens down.

$x^2=-4ay$, where $a=3$.

Therefore the equation is,

${\left(x\right)}^2=-4*3\left(y\right)$

$x^2=-12y$

The equation of the parabola is $x^2=-12y$.

To find the points that define the latus rectum, Let $y=-3$ in the above equation.

We get,

$x^2=-12*-3$

$x=\pm 6$

Hence $\left(6,-3\right)$ and $\left(-6,-3\right)$ defines the latus rectum.

To determine

c.

To graph: The parabola for the equation $x^2=-12y$.

Answer to Problem 21AYU

Explanation of Solution

Given:

Focus at $\left(0,-3\right)$; vertex at $\left(0,0\right)$.

Calculation:

The graph of the equation of the parabola $x^2=-12y$ is plotted below.

Using the given information $V\left(0,0\right)$ and $F\left(0,-3\right)$ we have formed the equation and plotted the graph for $x^2=-12y$. The focus $F\left(0,-3\right)$ and the vertex $V\left(0,0\right)$ lie on the horizontal line.

The equation for directrix D: $y=3$ is shown as dotted line. The parabola opens down. The points $\left(-6,-3\right)$ and $\left(6,-3\right)$ define the latus rectum.