Question

Find the standard form of the equation of the parabola with a focus at (0, -3) and a directrix at y = 3.

y2 = -12x

y2 = -3x

y = negative 1 divided by 12 x2

y = negative 1 divided by 3 x2

Answer 1

Option 3 : y = negative 1 divided by 12 x^2

The standard form of the equation of the parabola with a focus at (0, -3) and a directrix at y = 3 is
`y=(-1)/(12) x^(2)`

Solution:

Given that, parabola has a focus at (0, -3) and directrix at y = 3.

We have to find the parabola equation in standard form.

The standard form of parabola is given as:

`(x-h)^(2)=4 p(y-k)`

Where the focus is (h, k + p) and the directrix is y = k - p

So, here, (h, k + p) = (0, -3)

And y = k – p

y = 3

By comparison, h = 0, k + p = - 3 ------ eqn (2)

k – p = 3 ------ eqn (3)

Add (2) and (3)

k + p = -3

k – p = 3

(+)---------------

2k = 0

k = 0

Then, from (2) 0 + p = - 3

So, h = 0, k = 0, and p = -3

Now, put these values in (1)

`\begin{array}{l}{(x-h)^(2)=4 p(y-k)} \\\\ {(x-0)^(2)=4(-3)(y-0)} \\\\ {x^(2)=-12 y} \\\\{y=(-1)/(12) x^(2)}\end{array}`

So option 3 is correct