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 = \(11\over 4\)
