Question

What is the vertex form of the equation of the parabola with a focus at (200,-150) and a directrix of `y=135`?

Answer 1

The directrix is above the focus , so this is a parabola that opens downward .

Explanation:

The x-coordinate of the focus is also the x-coordinate of the vertex . So, we know that `h=200`.

Now, the y-coordinate of the vertex is halfway between the directrix and the focus:

`k=(1/2)[135+(-150)]=-15`

vertex `=(h,k)=(200,-15)`

The distance `p` between the directrix and the vertex is:

`p=135+15=150`

Vertex form : `(1/(4p))(x-h)^2+k`

Inserting the values from above into the vertex form and remember that this is downward opening parabola so the sign is negative :

`y=-(1/(4xx150))(x-200)^2-15`

`y=-(1/600)(x-200)^2-15`

Hope that helped