Courses
Courses for Kids
Free study material
Offline Centres
More
Store

# The mirror image of the directrix of the parabola ${y^2} = 4(x + 1)$ in the line mirror $x + 2y = 3$ is:1) x=-22) 4x-3y=163) 3x-4y+16=04) None of these

Last updated date: 20th Jun 2024
Total views: 375.3k
Views today: 7.75k
Verified
375.3k+ views
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.

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.