Question

What is the equation in standard form of the parabola with a focus at (12,-5) and a directrix of y= -6?

Answer 1

Because the directrix is a horizontal line, then the vertex form is `y = 1/(4f)(x - h)^2+k` where the vertex is `(h,k)` and f is the signed vertical distance from the vertex to the focus.

Explanation:

The the focal distance, f, is half of the vertical distance from the focus to the directrix:

`f = 1/2(-6--5)`

`f = -1/2`

`k = y_"focus"+f`

`k = -5 - 1/2`

`k = -5.5`

h is the same as the x coordinate of the focus

`h = x_"focus"`

`h = 12`

The vertex form of the equation is:

`y = 1/(4(-1/2))(x - 12)^2-5.5`

`y = 1/-2(x - 12)^2-5.5`

Expand the square:

`y = 1/-2(x^2 - 24x+ 144)-5.5`

Use the distributive property:

`y = -x^2/2 + 12x- 72-5.5`

Standard form:

`y = -1/2x^2 + 12x- 77.5`