# What is the equation of the parabola with a focus at (-15,-19) and a directrix of y= -8?

##### 2 Answers

#### Explanation:

Because the directrix is a horizontal line, we know that the parabola is vertically oriented (opens either up or down). Because the y coordinate of the focus (-19) below the directrix (-8), we know that the parabola opens down. The vertex form of the equation for this type of parabola is:

Where h is the x coordinate of the vertex, k it the y coordinated of the vertex, and the focal distance, f, is the half of the signed distance from directrix to the focus:

The y coordinate of the vertex, k, is f plus the y coordinate of the directrix:

The x coordinate of the vertex, h, is the same as the x coordinate of the focus:

Substituting these values into equation [1]:

Simplifying a bit:

#### Explanation:

Parabola is the locus of a point, which moves so that its distance from a line, called directix, and a point, called focus, are equal.

We know that the distance between two points

the distance between point

Now distance of a point

and its distance from directrix

Hence, equation of parabola would be

graph{x^2+30x+22y+522=0 [-56.5, 23.5, -35.28, 4.72]}