Given the focus and directrix of a parabola, how do we find the equation of the parabola?

If we consider only parabolas that open upwards or downwards, then the directrix will be a horizontal line of the form *y * = *c *.

Let (*a *, *b*) be the focus and let *y * = *c * be the directrix. Let (*x*_{0} , *y*_{0}) be any point on the parabola.

Any point, (*x*_{0} , *y*_{0}) on the parabola satisfies the definition of parabola, so there are two distances to calculate:

- Distance between the point on the parabola to the focus
- Distance between the point on the parabola to the directrix

To find the equation of the parabola, equate these two expressions and solve for *y*_{0} .

Find the equation of the parabola in the example above.

Distance between the point (*x*_{0} , *y*_{0}) and (*a *, *b*):

Distance between point ( *x*_{0} , *y*_{0}) and the line *y *= *c *:

(Here, the distance between the point and horizontal line is difference of their *y *-coordinates.)

Equate the two expressions.

Square both sides.

Expand the expression in *y*_{0} on both sides and simplify.

This equation in (*x*_{0} , *y*_{0}) is true for all other values on the parabola and hence we can rewrite with (*x *, *y*).

Therefore, the equation of the parabola with focus (*a *, *b*) and directrix *y * = *c * is

**Example**:

If the focus of a parabola is (2, 5) and the directrix is *y * = 3, find the equation of the parabola.

Let ( *x*_{0} , *y*_{0} ) be any point on the parabola. Find the distance between (*x*_{0} , *y*_{0}) and the focus. Then find the distance between (*x*_{0} , *y*_{0}) and directrix. Equate these two distance equations and the simplified equation in *x*_{0} and *y*_{0} is equation of the parabola.

The distance between (*x*_{0} , *y*_{0}) and (2, 5) is

The distance between (*x*_{0} , *y*_{0}) and the directrix, *y * = 3 is

| *y*_{0}– 3|.

Equate the two distance expressions and square on both sides.

Simplify and bring all terms to one side:

Write the equation with *y*_{0} on one side:

This equation in (*x*_{0} , *y*_{0}) is true for all other values on the parabola and hence we can rewrite with (*x *, *y*).

So, the equation of the parabola with focus (2, 5) and directrix is *y * = 3 is