# What is the standard form of the equation of the parabola with a directrix at x=-3 and a focus at (6,2)?

##### 1 Answer
Dec 31, 2017

The standard equation of horizontal parabola is
${\left(y - 2\right)}^{2} = 18 \left(x - 1.5\right)$

#### Explanation:

Focus is at $\left(6 , 2\right)$and directrix is $x = - 3$. Vertex is at midway

between focus and directrix. Therefore vertex is at

$\left(\frac{6 - 3}{2} , 2\right) \mathmr{and} \left(1.5 , 2\right)$.Here the directrix is at left of

the vertex , so parabola opens right and $p$ is positive.

The standard equation of horizontal parabola opening right is

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

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

vertex is $p = 6 - 1.5 = 4.5$. Thus the standard equation of

horizontal parabola is ${\left(y - 2\right)}^{2} = 4 \cdot 4.5 \left(x - 1.5\right)$ or

${\left(y - 2\right)}^{2} = 18 \left(x - 1.5\right)$

graph{(y-2)^2=18(x-1.5) [-40, 40, -20, 20]}