# How do you find the equation of the parabola having its focus at #( 1/2, 1/2)# and the directrix along #x - y = 1#?

##### 2 Answers

#### Explanation:

Calling the directrix as

and the symmetry line

we have

Solving for

Now, the parabola is the place where the distance between a generic plane point

but

where

giving

Attached the parabola plot

#### Explanation:

Let (x, y) be a point on the parabola.

Using that its distance from the focus equals the distance from the

directrix,

parabola. Squaring,

The vertex V of the parabola is on the perpendicular SX from the

focus S(1/2, 1/2), on the directrix DX given by x-y=1. The equation of

the perpendicular is of the form x+y=c. As S(1/2, 1/2) lies on this, c = 1.

V bisects SX.

So, V is (3/4, 1/4).. The size of the parabola

a = VS =