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.

Asked by Abhisek | 1 year ago |  177

##### Solution :- Consider the coordinates of point A as (a, 0)

Construct a line (AL) which is perpendicular to the x-axis

Here the angle of incidence is equal to angle of reflection

∠BAL = ∠CAL = Φ

∠CAX = θ

It can be written as

∠OAB = 180° – (θ + 2Φ) = 180° – [θ + 2(90° – θ)]

On further calculation

= 180° – θ – 180° + 2θ

= θ

So we get

∠BAX = 180° – θ

Slope of line = $$\dfrac{3-0}{5-a}$$

tanθ = $$\dfrac{3}{5-a}$$ ..............(1)

Slope of line AB = $$\dfrac{2-0}{1-a}$$

tanθ =(180° – θ) = $$\dfrac{2}{1-a}$$

- tanθ = $$\dfrac{2}{1-a}$$

- tanθ = $$\dfrac{2}{a-1}$$

From equations (1) and (2) ,

$$\dfrac{3}{5-a}$$$$\dfrac{2}{a-1}$$

By cross multiplication,
3a -3 =10-2a

$$\dfrac{13}{5}$$

Therefore, the coordinates of point A are $$( \dfrac{13}{5},0)$$

Answered by Pragya Singh | 1 year ago

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