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The mirror image of the directrix of the parabola ${y^2} = 4(x + 1)$ in the line mirror $x + 2y = 3$ is:
1) x=-2
2) 4x-3y=16
3) 3x-4y+16=0
4) None of these

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Last updated date: 20th Jun 2024
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Answer
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Hint: The above problem is based on the Parabola which has its standard equation as;
$y - k = 4a(x - h)$
where a is the distance from the vertex to focus, and the above equation is said to be parallel to x- axis.
Parabola is a plane curve which is approximately U shaped. It fits several other superficially different mathematical descriptions.
Using the above parabolic equation we will solve the given equation.

Complete step by step answer:
Let's define parabola in more detail and then we will do the calculation part of the problem.
Parabola involves a point(focus) and a line (directrix). Directrix is the line which is perpendicular to the axis of symmetry of a parabola and does not touch the parabola. Focus of a parabola is a fixed point on the interior of a parabola used in the formal definition of the centre.
Directrix of ${y^2} = 4(x + 1)$ is x= -2
Any point on x = -2 is (-2,k)
Now, mirror image (x, y) of (-2,k) in the line x + 2y = 3 is given by
$ \Rightarrow \dfrac{{x + 2}}{1} = \dfrac{{y - k}}{2} = - 2\left( {\dfrac{{ - 2 + 2k - 3}}{5}} \right)$ ..................1(Equation of the line which is mirror image of both x and y coordinates)
$ \Rightarrow x = \dfrac{{10 - 4k}}{5} - 2$ (for x coordinates)
$ \Rightarrow x = \dfrac{{ - 4k}}{5}$ .....................2
or
$ \Rightarrow k = \dfrac{{ - 5x}}{4}$ ..............2
Also, $y = \dfrac{{20 - 3k}}{5}$ ....................3(for y coordinates)
or
$ \Rightarrow y = 4 - \dfrac{{3k}}{5}$ ................3
From equation 2 and 3 we have substituted the value of x from equation 3.
$ \Rightarrow y = 4 + \left( {\dfrac{3}{5}} \right)\dfrac{{5x}}{4}$ ..............4
$ \Rightarrow 4y = 16 + 3x$
$ \Rightarrow 3x - 4y + 16 = 0$ (This is the required equation of the mirror image)

So, the correct answer is Option 3.

Note: Parabola has many applications such as a highway underpass is parabolic in shape, which is symmetric about a vertical line known as the axis of symmetry. Highway underpass is also parabolic in shape, the railway bridge over a road is in the shape of a parabola symmetric at the centre.