# jose's question at Yahoo! Answers: find the equation of the parabola given the focus and directrix

#### MarkFL

Staff member
Here is the question:

Writing an equation given the directrix and focus?

focus: (3,5) directrix y=1. write an equation for the parabola. How do I do this? please help!!
I have posted a link there to this thread so the OP can view my work.

#### MarkFL

Staff member
Re: jose's question at Yahoo! Questions: find the equation of the parabola given the focus and direc

Hello jose,

Let's let $(x,y)$ be an arbitrary point on the parabola. Now, we know the perpendicular distance from the point to the directrix will be equal to the distance between this point and the focus. Thus, we may state:

$$\displaystyle |y-1|=\sqrt{(x-3)^2+(y-5)^2}$$

Square both sides:

$$\displaystyle (y-1)^2=(x-3)^2+(y-5)^2$$

Expand binomials:

$$\displaystyle y^2-2y+1=x^2-6x+9+y^2-10y+25$$

Collect like terms:

$$\displaystyle 8y=x^2-6x+33$$

Divide through by $8$:

$$\displaystyle y=\frac{x^2-6x+33}{8}$$