Question

The equation of directrix of the parabola ${\left( {y - 2} \right)^2} = 4(x - 4)$
A) $x + 1 = 0$
B) $x = 1$
C) $x = 2$
D) $x = 5$

Answer 1

In the given question first convert it into the general form of the parabola that is ${y^2} = 4ax$ as we know the equation of this is $x = a$ so let us suppose that the $y - 2 = Y$ and $x - 4 = X$ equation become ${Y^2} = 4\left( 1 \right)X$ for this the directrix equation is $X = 1$

Complete step by step solution:
As in the given question we have to find out the equation of directrix of the parabola that is ${\left( {y - 2} \right)^2} = 4(x - 4)$
A parabola is the locus of the point in a plane which are an equal distance away from a given point and given line , the given line is known as the directrix of the parabola
In general if the parabola is ${y^2} = 4ax$ then the equation of directrix is $x = a$ So the above equation we to convert it into general form ,
So let us suppose that the $y - 2 = Y$ and $x - 4 = X$ now put it in into the given equation that is ${\left( {y - 2} \right)^2} = 4(x - 4)$
It looks as ${Y^2} = 4X$ or ${Y^2} = 4\left( 1 \right)X$
Hence on comparing with general equation that is ${y^2} = 4ax$ we get $a = 1$
So the equation of the directrix of the ${Y^2} = 4X$ is $X = 1$
We know that the $x - 4 = X$ as it is suppose by us now put it in the equation of directrix ,
$x - 4 = 1$
on solving we get $x = 5$ hence it is the required equation of the directrix .

So, the correct answer is “Option D”.

Note: As for the parabola ${y^2} = 4ax$ the axis is x- axis and the y- axis for the given question ${\left( {y - 2} \right)^2} = 4(x - 4)$ the axis become $y = 2$ and $x = 4$ it is shifted parabola and the focus of this parabola is at the coordinate $\left( {5,2} \right)$