Question

A ray of light is coming along the line y = b from the positive direction of x-axis and strikes a concave mirror whose intersection with the x-y plane is a parabola y² = 4ax. Find the equation of the reflected ray and show that it passes through the focus of the parabola. Both a and b are positive. 

Answer 1

A ray of light along y = b intersects the parabola at p(b²/4a ,b) 

Equation to the parabola is y² = 4ax  ....(1)

Differentiating (1) dy/dx = 2a/y

Slope of normal at P is - b/2a

 If the angle between ray of light and the normal is θ then tan θ = b/2a

Let PQ be reflected ray

Angle between PQ and the normal at P is also θ

Let the slope of PQ be m

4abx + (5a² – b²) y – 4a² b = 0

Substitute S(a, 0)

Reflected ray passes through the focus.