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

Nov 22, 2015

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:

$k = \left(\frac{1}{2}\right) \left[135 + \left(- 150\right)\right] = - 15$

vertex $= \left(h , k\right) = \left(200 , - 15\right)$

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

$p = 135 + 15 = 150$

Vertex form : $\left(\frac{1}{4 p}\right) {\left(x - h\right)}^{2} + k$

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

$y = - \left(\frac{1}{4 \times 150}\right) {\left(x - 200\right)}^{2} - 15$

$y = - \left(\frac{1}{600}\right) {\left(x - 200\right)}^{2} - 15$

Hope that helped