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.

$x^2=-4y$

Short answer

a. The parabola $x^2=-4y$has its focus at $\left(0,-1\right)$. Its focal diameter is 4 and directrix is $y=1$.

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

Step-by-step solution

a.Step 1. Given information.

The given equation of a parabola $x^2=-4y$.

Step 2. Write the concept.

$x^2=4(-1)y$The given equation of a parabola $x^2=-4y$

Putting the equation in standard form $x^2=4ay$

Thus we get, $x^2=-4y$

$x^2=4(-1)y$

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

By solving $a=-1$ for a,

We get $a=-1$

$\therefore$ the focus is $\left(0,-1\right)$ and the directrix is $y=1$

b.Step 1. Given information.

The given equation of a parabola $x^2=-4y$.

Step 2. Write the concept.

The given equation of a parabola $x^2=-4y$.

Converting the equation in standard form

Thus, $x^2=-4y$ which represents a downward facing parabola with vertex at (0,0)

$x^2=-4y$

Hence the coordinates of the focus will be $\left(0,-1\right)$

And the equation of the directrix will be $y=1$