Starting at the origin, a beam of light hits a mirror (in the form of a straight line) at point A = (4, 8) and is reflected to point B = (8, 12). Compute the exact slope of the mirror.

Slope of the incident ray = (8-0)/(4-0) = 2 ( angle A)

Slope of the reflected ray = (12-8)/(8-4) = 1( angle B)

Now let us find the line that bisects the 2 rays

Tan (A+B) = (tan A + tan B)/(1- Tan A tan B) = (1+2)/(1-2) = - 3

Let C be the angle bisector between A and B

A + B = 2C say m is slope of the bisector

So we get 2m/(1-m^2) = - 3

Or 2m = -3 + 3m^2
Or 3m^2 – 2m – 3 = 0
=> m = (1/6) [2 + √(4 + 36)] ... [m positive taken to the correct slope
=> m = (1/3) [1 + √(10)].

this is the slope of the mirror

The reason for the slope to be positive is that from 4,8 one line goes to (8,12) that is line is pointing towards 1st quadrant and (0,0) is to the direction of 3rd quadrant. so the tangent should have a slope between 1 and 2 and not the normal