# What is the equation of the parabola with a focus at (15,-3) and a directrix of y= -4?

May 18, 2016

Equation of parabola is ${x}^{2} - 30 x - 2 y + 218 = 0$

#### Explanation:

Here the directrix is a horizontal line $y = - 4$.

Since this line is perpendicular to the axis of symmetry, this is a regular parabola, where the $x$ part is squared.

Now the distance of a point on parabola from focus at $\left(15 , - 3\right)$ is always equal to its between the vertex and the directrix should always be equal. Let this point be $\left(x , y\right)$.

Its distance from focus is $\sqrt{{\left(x - 15\right)}^{2} + {\left(y + 3\right)}^{2}}$ and from directrix will be $| y + 4 |$

Hence, ${\left(x - 15\right)}^{2} + {\left(y + 3\right)}^{2} = {\left(y + 4\right)}^{2}$

or ${x}^{2} - 30 x + 225 + {y}^{2} + 6 y + 9 = {y}^{2} + 8 y + 16$

or ${x}^{2} - 30 x - 2 y + 234 - 16 = 0$

or ${x}^{2} - 30 x - 2 y + 218 = 0$