Question

Find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand .

Focus at $\left(-3,-2\right)$; directrix the line x = 1

Short answer

The equation of the parabola is $y^2=8x$and the points that determine latus rectum are $\left(2,\frac{1}{2}\right),\left(-2,\frac{1}{2}\right)$.

The graph of the equation shown below .

Step-by-step solution

Step 1.  Given information .

Consider the given focus and the line of directrix .

Step 2. Equation of parabola used .

A parabola whose focus is at $\left(-3,-2\right)$and whose directrix line is $x=1$has its vertex $\left(-1,-2\right)$. Since the focus is on the negative y axis then the equation of parabola is of the form$y^2=-4ax$.

Step 3.  Find the equation of parabola .

To find the equation of parabola substitute the values in the equation .

Here $a=-2$

$$\begin{gathered} y^2=-4ax \\ {y}^2=-4·-2x \\ {y}^2=8x \end{gathered}$$

Further simplify .

role="math" localid="1647157860145" $$Substitute $y=-2$in the equation .

$$\begin{gathered} y^2=-4ax \\ {\left(-2\right)}^2=-4·-2x \\ x=\frac{1}{2} \end{gathered}$$

Further simplify.

$$\begin{gathered} y^2=-4ax \\ {y}^2=-4·-2·\frac{1}{2} \\ {y}^2=\pm 2 \end{gathered}$$

Therefore the two points that define latus rectum are$\left(2,\frac{1}{2}\right),\left(-2,\frac{1}{2}\right)$.

Step 4.  Plot the graph .

The graph of the equation $y^2=8x$is shown below .