Chapter 10.2, Problem 24AYU

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 $\left(0,-1\right)$; directrix the line $y\text{ = 1}$

To determine

(a & b)

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

Focus at $\left(0,-1\right)$; directrix the line $y\text{ = 1}$.

Answer to Problem 24AYU

$x^2=-4y$, Latus rectum: $\left(2,-1\right)$ and $\left(-2,-1\right)$.

Explanation of Solution

Given:

Focus at $\left(0,-1\right)$; directrix at the line $y\text{ = 1}$.

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:

Given that the focus is at $F\left(0,-1\right)$ and the directrix is the line $y\text{ = 1}$. The vertex is $\left(0,0\right)$. Both vertex $V\left(0,0\right)$ and $F\left(0,-1\right)$ lies on the vertical line $x=\text{ 0}$ (The axis of symmetry). The distance $a$ from the vertex $V\left(0,0\right)$ to the focus $F\left(0,-1\right)$ is $a=\text{ 1}$.

Also, because the focus $F\left(0,-1\right)$ lies down to the vertex, the parabola opens down.

$x^2=-4ay$

$a=\text{ 1}$

Therefore the equation is,

$x^2=-4*\text{ 1 }*y$

$x^2=-4y$

The equation of the parabola is $x^2=-4y$.

To find the points that define the latus rectum, Let $y\text{ = 1}$ in the above equation.

We get,

$x^2=-4*\text{ 1}$

$x=\pm 2$

Hence $\left(2,-1\right)$ and $\left(-2,-1\right)$ defines the latus rectum.

To determine

c.

To graph: The parabola for the equation $x^2=-4y$.

Answer to Problem 24AYU

Explanation of Solution

Given:

Focus at $\left(0,-1\right)$; vertex at $\left(0,0\right)$.

Calculation:

The graph of the parabola for the equation $x^2=-4y$.

Using the given information D: $y\text{ = 1}$ and $F\left(0,-1\right)$ we have formed the equation and plotted the graph for $x^2=-4y$. The focus $F\left(0,-1\right)$ and the vertex $V\left(0,0\right)$ lie on the horizontal line.

The equation for directrix D: $y\text{ = 1}$ is shown as dotted line. The parabola opens down. The points $\left(2,-1\right)$ and $\left(-2,-1\right)$ define the latus rectum.