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Question: The mirror image of the directrix of the parabola \({y^2} = 4(x + 1)\) in the line mirror \(x + 2y =...

The mirror image of the directrix of the parabola y2=4(x+1){y^2} = 4(x + 1) in the line mirror x+2y=3x + 2y = 3 is:

  1. x=-2
  2. 4x-3y=16
  3. 3x-4y+16=0
  4. None of these
Explanation

Solution

The above problem is based on the Parabola which has its standard equation as;
y−k=4a(x−h)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 y2=4(x+1){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
⇒x+21=y−k2=−2(−2+2k−35)\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)
⇒x=10−4k5−2\Rightarrow x = \dfrac{{10 - 4k}}{5} - 2 (for x coordinates)
⇒x=−4k5\Rightarrow x = \dfrac{{ - 4k}}{5} .....................2
or
⇒k=−5x4\Rightarrow k = \dfrac{{ - 5x}}{4} ..............2
Also, y=20−3k5y = \dfrac{{20 - 3k}}{5} ....................3(for y coordinates)
or
⇒y=4−3k5\Rightarrow y = 4 - \dfrac{{3k}}{5} ................3
From equation 2 and 3 we have substituted the value of x from equation 3.
⇒y=4+(35)5x4\Rightarrow y = 4 + \left( {\dfrac{3}{5}} \right)\dfrac{{5x}}{4} ..............4
⇒4y=16+3x\Rightarrow 4y = 16 + 3x
⇒3x−4y+16=0\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.