# What is the standard form of the equation of the parabola with a directrix at x=4 and a focus at (-7,-5)?

Dec 25, 2017

The standard equation of parabola is ${\left(y + 5.5\right)}^{2} = - 22 \left(x + 1.5\right)$

#### Explanation:

Focus is at $\left(- 7 , - 5\right)$ and directrix is $x = 4$. Vertex is at midway

between focus and directrix. Therefore vertex is at

$\left(\frac{- 7 + 4}{2} , - 5\right) \mathmr{and} \left(- 1.5 , - 5\right)$ The equation of horizontal

parabola opening left is

(y-k)^2 = -4p(x-h) ; h=-1.5 ,k=-5

or ${\left(y + 5.5\right)}^{2} = - 4 p \left(x + 1.5\right)$ . The distance between focus and

vertex is $p = 7 - 1.5 = 5.5$. Thus the standard equation of

horizontal parabola is ${\left(y + 5.5\right)}^{2} = - 4 \cdot 5.5 \left(x + 1.5\right)$ or

${\left(y + 5.5\right)}^{2} = - 22 \left(x + 1.5\right)$

graph{(y+5.5)^2=-22(x+1.5) [-160, 160, -80, 80]}