# What is the equation of the parabola with a focus at (2,15) and a directrix of y= -25?

Oct 23, 2017

The equation of parabola is $y = \frac{1}{20} {\left(x - 2\right)}^{2} - 5$

#### Explanation:

Focus is at $\left(2 , 15\right)$and directrix is $y = - 25$. Vertex is at midway

between focus and directrix. Therefore vertex is at $\left(2 , \frac{15 - 25}{2}\right)$

or at $\left(2 , - 5\right)$ . The vertex form of equation of parabola is

y=a(x-h)^2+k ; (h.k) ; being vertex. $h = 2 \mathmr{and} k = - 5$

So the equation of parabola is $y = a {\left(x - 2\right)}^{2} - 5$. Distance of

vertex from directrix is $d = 25 - 5 = 20$, we know $d = \frac{1}{4 | a |}$

$\therefore 20 = \frac{1}{4 | a |} \mathmr{and} | a | = \frac{1}{20 \cdot 4} = \frac{1}{80}$. Here the directrix is behind

the vertex , so parabola opens upward and $a$ is positive.

$\therefore a = \frac{1}{80}$ . The equation of parabola is $y = \frac{1}{20} {\left(x - 2\right)}^{2} - 5$

graph{1/20(x-2)^2-5 [-40, 40, -20, 20]} [Ans]