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

Solution:

We can see the scenario of the given problem in the following figure.

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 and ∠CAX = θ

Now,

∠OAB = 180° - (θ + 2Φ)

= 180° - [θ + 2 (90° - θ)]

= 180° - [θ + 180° - 2θ]

= 180° - 180° + θ

= θ

Therefore,

∠BAX = 180° - θ

Now,

Slope of line AC = (3 - 0)/(5 - a)

⇒ tanθ = 3/(5 - a) ....(1)

⇒ Slope of line AB = (2 - 0)/(1 - a)

⇒ tan (180° - θ) = 2/(1 - a)

⇒ tanθ = 2/(1 - a) ....(2)

From equations (1) and (2), we obtain

⇒ 3/(5 - a) = 2/(1 - a)

⇒ 3a - 3 = 10 - 2a

⇒ a = 13/5

Thus, the coordinates of point A are (13/5, 0)

---

NCERT Solutions Class 11 Maths Chapter 10 Exercise ME Question 22

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

Summary:

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). Then the coordinates of A are (13/5, 0)