Question

A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

Answer 1

Let AB be the line joining the point and the image such that it is perpendicular to the x-axis and the x-axis bisects the line.

B(1,2)
C(5,3)

A=?

Let A be (x,y)

Therefore, (5,3)=[({1+x}/2), ({2+y}/2)]       (Mid point formula)

If the coordinates in LHS and RHS are equal, their respective x-coordinates and y- coordinates are also equal.
Therefore, 5=(1+x)/2.....(1) & 3=(2+y)/2.......(2)

(1)=> 10=1+x      (2)=> 6=2+y
           x=9                     y=4

Therefore, the image of the point (1,2) lies on (9,4).


 

Answer 2

Let the coordinates of point A be (a, 0). Draw a line (AL) perpendicular to the x-axis. We know that angle of incidence is equal to angle of reflection. Hence, let ∠BAL = ∠CAL = Φ Let ∠CAX = θ ∴∠OAB = 180° – (θ + 2Φ) = 180° – [θ + 2(90° – θ)] = 180° – θ – 180° + 2θ = θ ∴ ∠BAX = 180° – θ Read more on Sarthaks.com - https://www.sarthaks.com/33082/light-passing-through-point-reflects-axis-point-and-the-reflected-ray-passes-through-point