The equation of the directrix of the parabola y² + 4x + 4y + 2 is

Concept:

A parabola refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed-line. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola. Also, an important point to note is that the fixed point does not lie on the fixed-line. A locus of any point which is equidistant from a given point (focus) and a given line (directrix) is called a parabola. 

The standard equation of a regular parabola is y2 = 4ax.

Here,

• Coordinates of vertex: (0, 0)
• Coordinates of focus: (a, 0)
• Equation of the directrix: x = -a
• Equation of axis: y = 0
• Length of the latus rectum: 4a

• Focal distance of a point P(x, y): a + x

Formula used:

(a + b)2 = a2 + b2 + 2ab

Calculation:

Given equation of the parabola is-

y2 + 4x + 4y + 2 = 0

⇒ (y2 + 4y + 4) + 4x - 2 = 0

⇒ (y + 2)2 = -4x + 2      (∵ (a + b)2 = a2 + b2 + 2ab)

\(⇒ (y + 2)^2 = -4\left (x - \frac{1}{2} \right )\)

Let y + 2 = Y and (x - 1/2) = X

⇒ Y2 = -4X

On comparing this equation with y2 = -4ax, we get

a = 1

⇒ Equation of directrix = (X = a)

\(⇒ \left (x - \frac{1}{2} \right ) = 1\)

\(\Rightarrow x=\dfrac{3}{2}\)

Hence, The equation of the directrix of the parabola y2 + 4x + 4y + 2 = 0 is \(x=\dfrac{3}{2}\).