# What is the vertex form of the equation of the parabola with a focus at (-4,-7) and a directrix of y=10?

May 7, 2018

The equation of parabola is $y = - \frac{1}{34} {\left(x + 4\right)}^{2} + 1.5$

#### Explanation:

Focus is at $\left(- 4 , - 7\right)$and directrix is $y = 10$. Vertex is at midway

between focus and directrix. Therefore vertex is at

$\left(- 4 , \frac{10 - 7}{2}\right) \mathmr{and} \left(- 4 , 1.5\right)$ . The vertex form of equation of

parabola is y=a(x-h)^2+k ; (h.k) ; being vertex.

$h = - 4 \mathmr{and} k = 1.5$. So the equation of parabola is

$y = a {\left(x + 4\right)}^{2} + 1.5$. Distance of vertex from directrix is

$d = 10 - 1.5 = 8.5$, we know $d = \frac{1}{4 | a |}$

$\therefore 8.5 = \frac{1}{4 | a |} \mathmr{and} | a | = \frac{1}{8.5 \cdot 4} = \frac{1}{34}$. Here the directrix is

above the vertex , so parabola opens downward and $a$ is

negative $\therefore a = - \frac{1}{34}$ Hence the equation of parabola is

$y = - \frac{1}{34} {\left(x + 4\right)}^{2} + 1.5$

graph{-1/34(x+4)^2+1.5 [-40, 40, -20, 20]}