please help me to solve this assignment :

consider a 2D mirror has a parabolic profile of y = (x-3)^2 + 5 facing toward y^+ direction. Prove that any beam of light in y^- direction that this hits the mirror will be reflected toward a particular point (i.e. known as the focus of parabolic mirror.

To prove that any beam of light in the y^- direction hits the mirror will be reflected toward the focus of the parabolic mirror, we need to show that the incident angle is equal to the angle of reflection.

Let's break down the problem into steps:

Step 1: Understand the mirror's profile
The given mirror has a parabolic profile expressed by the equation y = (x-3)^2 + 5. This equation represents a vertical parabola with its vertex at (3, 5). The vertex is the lowest point on the parabola, also known as the focus.

Step 2: Determine the slope of the mirror at any point
To understand how the light beam will be reflected by the mirror, we need to find the slope of the mirror at the point of contact. This slope will help us determine the angle of incidence and reflection.

Differentiating the equation of the parabola with respect to x, we can find the slope of the mirror at any point:
dy/dx = 2(x-3)

Step 3: Identify the point of contact
Let's assume the point of contact between the light beam and the mirror is (x, y). The tangent line to the mirror at this point would have a slope equal to dy/dx evaluated at that point.

Step 4: Find the incident angle
The incident angle, denoted as θ_i, is the angle between the normal (the line perpendicular to the mirror's surface at the point of contact) and the direction of the incoming light beam. Since the mirror is symmetrical, the normal is also perpendicular to the tangent line.

Step 5: Find the angle of reflection
The angle of reflection, denoted as θ_r, is the angle between the normal and the direction of the reflected light beam.

Step 6: Prove that the incident angle equals the angle of reflection
To prove that the light beam is reflected toward the focus of the parabolic mirror, we need to show that θ_i = θ_r.

Now, let's proceed with solving the problem based on these steps.