Directrix of Parabola – Equation and Formula

Here you will learn formula for finding the equation of directrix of parabola with examples.

Let’s begin –

Equation of Directrix of Parabola

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

The equation of directrix is x = -a.

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

The equation of directrix is x = a.

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

The equation of directrix is y = -a

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

The equation of directrix is y = a.

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

The equation of directrix is x + a – h = 0.

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

The equation of directrix is y + a – q = 0.

Also Read : Different Types of Parabola Equations

Example : For the given parabola, find the equation of the directrices :

(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 equation of the directrix is x = -a i.e. x = -2.

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

Hence, the equation of the directrix is y = a i.e. y = 4.

Example : Find the equation of directrix 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 equation of directrix is x + a – h = 0

i.e.  x + (1/4) – 3 = 0 $\implies$ x = $11\over 4$