Question:

The mirror image of any point on the directrix of $ y^2 = 4(x + 1)$ lies on

Updated On: Jul 7, 2022
  • $3x + 4y- 16 = 0 $
  • $3x - 4y +16 = 0 $
  • $3x + 4y + 16 = 0 $
  • $3x - 4y - 16 = 0.$
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The Correct Option is B

Solution and Explanation

Directrix of $y^{2}= 4\left(x+1\right)$ is $x=-2$ Any point on it is $\left(-2, k\right)$ Now mirror image (x, y) of (- 2, k) in the line x + 2y = 3 is given by $Also y = k +4 -\frac{8k}{5}$ y = k +4 - $\frac{8k}{5}$ Hence y = 4 - $\frac{3k}{5}=4+\frac{3}{5}\left(\frac {5x}{4}\right)=\frac {16+3x}{4}$ $\Rightarrow\,\,$3x -4y + 16 = 0 is the reqd. mirror image.
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Concepts Used:

Parabola

Parabola is defined as the locus of points equidistant from a fixed point (called focus) and a fixed-line (called directrix).

Parabola


 

 

 

 

 

 

 

 

 

Standard Equation of a Parabola

For horizontal parabola

  • Let us consider
  • Origin (0,0) as the parabola's vertex A,
  1. Two equidistant points S(a,0) as focus, and Z(- a,0) as a directrix point,
  2. P(x,y) as the moving point.
  • Let us now draw SZ perpendicular from S to the directrix. Then, SZ will be the axis of the parabola.
  • The centre point of SZ i.e. A will now lie on the locus of P, i.e. AS = AZ.
  • The x-axis will be along the line AS, and the y-axis will be along the perpendicular to AS at A, as in the figure.
  • By definition PM = PS

=> MP2 = PS2 

  • So, (a + x)2 = (x - a)2 + y2.
  • Hence, we can get the equation of horizontal parabola as y2 = 4ax.

For vertical parabola

  • Let us consider
  • Origin (0,0) as the parabola's vertex A
  1. Two equidistant points, S(0,b) as focus and Z(0, -b) as a directrix point
  2. P(x,y) as any moving point
  • Let us now draw a perpendicular SZ from S to the directrix.
  • Then SZ will be the axis of the parabola. Now, the midpoint of SZ i.e. A, will lie on P’s locus i.e. AS=AZ.
  • The y-axis will be along the line AS, and the x-axis will be perpendicular to AS at A, as shown in the figure.
  • By definition PM = PS

=> MP2 = PS2

So, (b + y)2 = (y - b)2 + x2

  • As a result, the vertical parabola equation is x2= 4by.