Question

The equation of directrix of the parabola ${\left( {y - 2} \right)^2} = 4\left( {x - 4} \right)$, is
A) $x + 1 = 0$
B) $x = 1$
C) $x = 2$
D) $x = 4$

Answer 1

Here, we are required to find the equation of directrix of the given parabola. We will compare the given equation of the parabola with the standard equation of parabola and hence, find the value of the unknown variable. Then, we will observe in which direction the given parabola will open and according to that, we will use the formula of the equation of directrix. Then we will substitute the value of the obtained variable to find the required equation of directrix of the given parabola.

Formula Used:
We will use following formulas:

1. Equation of a parabola is ${y^2} = 4ax$
2. Equation of the directrix is $x = - a$ where the parabola opens toward the right and $a$ is a constant.

Complete step by step solution:
Given equation of the parabola is: ${\left( {y - 2} \right)^2} = 4\left( {x - 4} \right)$
Now, we know that the equation of a parabola is of the form ${Y^2} = 4aX$.
Here, comparing the given equation, i.e. ${\left( {y - 2} \right)^2} = 4\left( {x - 4} \right)$ with the standard equation of the parabola, we get,
$Y = y + 2$ and $X = x - 4$
Hence, substituting these values in the given equation ${\left( {y - 2} \right)^2} = 4\left( {x - 4} \right)$ , we get,
${Y^2} = 4X$
Now, since, the equation of a parabola is of the form ${Y^2} = 4aX$.
Hence, comparing ${Y^2} = 4aX$ with ${Y^2} = 4X$ here, $a = 1$
Hence, the equation of the directrix will be of the form $x = - a$ (as this parabola opens towards the right).
Here, in this question, $a = 1$
Therefore, equation of the directrix will be:
$x = - 1$
Adding 1 on both sides,
$\Rightarrow x + 1 = 0$
Hence, the equation of directrix of the parabola ${\left( {y - 2} \right)^2} = 4\left( {x - 4} \right)$ is $x + 1 = 0$

Therefore, option A is the correct answer.

Note:
A parabola is a plain curve which is mirror-symmetrical and U-shaped. Also, every point on a parabola is at equal distance from the focus i.e. the fixed point. Now, a directrix is a line perpendicular to the axis of symmetry of the parabola. Also, as the parabola opens towards the right, so, we have taken the equation of the directrix of the form $x = - a$. If the parabola would be opening towards the upward or downward direction, we would have taken the equation of the directrix as $x = a$. Hence, we should know the difference between these two formulas of equation of directrix of a parabola to answer this question.