Question

An equation of a parabola is

given.

(a) Find the focus, directrix, and focal diameter of the

parabola.

(b) Sketch a graph of the parabola and its directrix.

$y^2=-24x$

Short answer

a. The parabola $y^2=-24x$ has its focus at $\left(-6,0\right)$ . Its focal diameter is $24$ and directrix is $x=6$.

b. The graph of parabola $y^2=-24x$is

Step-by-step solution

a.Step 1. Given information.

The given equation of a parabola $y^2=-24x$.

Step 2. Write the concept.

The given equation of a parabola $y^2=-24x$

Putting the equation in standard form $y^2=4ax$

Thus we get, $y^2=-24x$

$y^2=4(-6)x$

From the above equation $4a=-24$, so the focal diameter is 8

By solving $a=-6$ for a,

We get $a=-6$

$\therefore$the focus is $\left(-6,0\right)$and the directrix is x=6

b.Step 1. Given information.

The given equation of a parabola $y^2=-24x$.

Step 2. Write the concept.

The given equation of a parabola $y^2=-24x$.

Converting the equation in standard form

Thus, $y^2=-24x$ which represents a right facing parabola with vertex at (0,0)

$y^2=-24x$

Hence the coordinates of the focus will be $\left(-6,0\right)$

And the equation of the directrix will be $x=6$