Question

What is the equation in standard form of the parabola with a focus at (2,-4) and a directrix of y= 6?

Answer 1

`y = -1/20x^2 + 1/5x + 4/5`

Explanation:

The general form for the equation of a horizontal directrix is `y = k - f`

The focus of a parabola with a horizontal directrix is of the general form `(h, k + f)`

Therefore, we can write these 3 equations that will help us:

h = 2

-4 = k + f

6 = k - f

solve the last two equation for k and f:

`k = 1`
`f = -5`

The vertex form of the equation of this type of parabola is:

`y = a(x - h)^2 + k`

Because we are not given the value of a, substitute `1/(4f)` for `a`:

`y = 1/(4f)(x - h)^2 + k`

Substitute our known values into the above equation:

`y = 1/(4(-5))(x - 2)^2 + 1`

Expand the square:

`y = -1/20(x^2 - 4x+ 4) + 1`

Distribute `-1/20`:

`y = -1/20x^2 + 1/5x - 1/5 + 1`

Combine the constants:

`y = -1/20x^2 + 1/5x + 4/5`