Question

How do you write the equation given focus (-4,-2) directrix x=-8?

Answer 1

Please see the explanation.

Explanation:

The x coordinate of the vertex `x_v` is halfway between the x coordinate of the directrix, -8, and the x coordinate of the focus,-4:

`x_v =-4 + (-8 - -4)/2`

`x _v = -4 -2`

`x _v = -6`

The y coordinate of the vertex is the same as the focus, therefore, the vertex is: `(-6, -2)`

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

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

where `(h,k)` is the vertex.

Substitute the vertex into the equation:

`x = a(y - -2)^2 - 6`

To find the value of "a", use the fact that the distance, f, between the vertex and the focus is 2:

`a = 1/(4f)`

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

`a = 1/8`

The equation is:

`x = 1/8(y - -2)^2 - 6`