# Focus of Parabola Coordinates with Examples

Here you will learn how to find the focus of parabola with examples.

Let’s begin –

## Focus of Parabola Coordinates

#### (i) For Parabola $$y^2$$ = 4ax :

The coordinates of focus is (a, 0).

#### (ii) For Parabola $$y^2$$ = -4ax :

The coordinates of focus is (-a, 0).

#### (iii) For Parabola $$x^2$$ = 4ay :

The coordinates of focus is (0, a).

#### (iv) For Parabola $$x^2$$ = -4ay :

The coordinates of focus is (0, -a).

#### (v) For Parabola $$(y – k)^2$$ = 4a(x – h) :

The coordinates of focus is (h + a, k).

#### (vi) For Parabola $$(x – p)^2$$ = 4a(y – q) :

The coordinates of focus is (p, a + q).

Also Read : Different Types of Parabola Equations

Example : For the given parabola, find the coordinates of the foci :

(i) $$y^2$$ = 8x

(ii) $$x^2$$ = -16y

Solution :

(i) The given parabola is of the form $$y^2$$ = 4ax, where 4a = 8 i.e. a = 2.

Hence, the coordinates of the focus are (a, 0) i.e. (2, 0).

(ii) The given parabola is of the form $$x^2$$ = -4ay, where 4a = 16 i.e. a = 4.

Hence, the coordinates of the focus are (0, -a) i.e. (0, -4).

Example : Find the coordinates of foci of the parabola $$y^2 – 8y – x + 19$$ = 0

Solution : The given equation is

$$y^2 – 8y – x + 19$$ = 0  $$\implies$$  $$y^2 – 8y$$ = x – 19

$$\implies$$  $$y^2 – 8y + 16$$ = x – 19 + 16

$$\implies$$  $$(y – 4)^2$$ = (x – 3)

The equation is of the form $$(y – k)^2$$ = 4a(x – h),

On Comparing we get,

4a = 1 i.e. a = 1/4 and k = 4, h = 3

Hence, the coordinates of focus is (h + a, k).

i.e.  ((3 + 1/4), 4) = (13/4, 4).