# What is the standard form of the equation of the parabola with a directrix at x=3 and a focus at (-5,-5)?

Jun 2, 2018

The equation of parabola is ${\left(y + 5\right)}^{2} = - 16 \left(x + 1\right)$

#### Explanation:

Focus is at $\left(- 5 , - 5\right)$and directrix is $x = 3$. Vertex is at midway

between focus and directrix. Therefore vertex is at

$\left(\frac{- 5 + 3}{2} , - 5\right) \mathmr{and} \left(- 1 , - 5\right)$ The directrix is at the right side

of vertex ,so, the horizontal parabola opens left. The equation of

horizontal parabola opening left is ${\left(y - k\right)}^{2} = - 4 p \left(x - h\right)$

$h = - 1 , k = - 5$ or ${\left(y + 5\right)}^{2} = - 4 p \left(x + 1\right)$ . the distance

between focus and vertex is $p = 5 - 1 = 4$. Thus the standard

equation of horizontal parabola is ${\left(y + 5\right)}^{2} = - 4 \cdot 4 \left(x + 1\right)$

or ${\left(y + 5\right)}^{2} = - 16 \left(x + 1\right)$

graph{(y+5)^2= -16(x+1) [-80, 80, -40, 40]} [Ans]