Answer
Verified
407.1k+ 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.
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.
$y - k = 4a(x - h)$
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Which are the Top 10 Largest Countries of the World?
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE