Question

What is the equation of the parabola with a focus at (-3,4) and a directrix of y= -7?

Answer 1

`y = 1/22(x- (-3))^2 -3/2`

Explanation:

Because the directrix is a horizontal line, we know that the vertex has the same x coordinate as the focus and has a y coordinate that is the midpoint between the directrix and the focus:

`(h,k) = (-3,(4+ (-7))/2)`

`(h,k) = (-3,-3/2)`

Substitute the values for h and k into the vertex form for the equation of a parabola that opens vertically:

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

`y = a(x- (-3))^2 -3/2`

Compute the signed vertical distance, `f`, from the vertex to the focus:

`f = 4 - (-3/2)`

`f = 11/2`

We know that `a = 1/(4f)`

`a = 1/(4(11/2))`

`a = 2/(4(11))`

`a = 1/22`

Substitute the value for "a" into the equation:

`y = 1/22(x- (-3))^2 -3/2`