# What is the equation of the parabola with a focus at (7,3) and a directrix of y= 4?

Aug 15, 2017

The equation of parabola is $y = - 0.5 {\left(x - 7\right)}^{2} + 3.5$

#### Explanation:

Focus is at $\left(7 , 3\right)$ , Directrix is $y = 4$ . The vertex is at midway

between focus and directrix .

So vertex is at $\left(7 , \frac{3 + 4}{2}\right) \mathmr{and} \left(7 , 3.5\right)$ , The distance between

vertex and directrix is $d = 4 - 3.5 = 0.5$ . Here directrix is above

vertex , so parabola opens downward and $a$ is negative.

$d = \frac{1}{4 | a |} \mathmr{and} 0.5 = \frac{1}{4 | a |} \therefore | a | = \frac{1}{2} , a = - 0.5$

The equation of parabola is y= a(x-h)^2+k ; (h,k)

being vertex. Here $h = 7 , k = 3.5 , a = - 0.5$ . Hence

the equation of parabola is $y = - 0.5 {\left(x - 7\right)}^{2} + 3.5$

graph{-0.5(x-7)^2 +3.5 [-20, 20, -10, 10]}