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 hitting the mirror from the y^- direction will be reflected toward the focus of the parabolic mirror, we can use the principle of reflection and the properties of the parabola.
The general equation of a parabola is y = ax^2 + bx + c, where a, b, and c are constants. Given that the mirror has a parabolic profile of y = (x-3)^2 + 5, we can rewrite it as y = x^2 - 6x + 14.
Let's assume a beam of light traveling in the y^- direction, meaning it will hit the mirror with an angle of incidence that is normal to the mirror surface. The angle of incidence (θi) is defined as the angle between the incoming beam and the normal to the mirror surface.
Now, consider a point P on the mirror surface, with coordinates (x, y). When the beam of light hits point P, it will be reflected according to the law of reflection: the angle of incidence is equal to the angle of reflection (θi = θr), and both angles are measured with respect to the normal to the mirror surface.
Since the beam is approaching from the y^- direction, the angle of incidence (θi) is 90 degrees. Therefore, the angle of reflection (θr) will also be 90 degrees. This implies that the reflected beam will propagate parallel to the x-axis.
Now, to determine the point towards which the reflected beam converges, we need to find the point where all the reflected beams intersect. This point is known as the focus of the parabolic mirror.
For a parabolic mirror, the focus is located at the point (p, q), where p = -b/2a. In our case, the equation of the mirror is y = x^2 - 6x + 14, so a = 1, b = -6, and c = 14. Plugging these values into the formula, we can find the x-coordinate of the focus:
p = -(-6)/2(1) = 3
Therefore, the x-coordinate of the focus is 3. To find the y-coordinate, we substitute this value of x back into the equation of the mirror:
y = (3-3)^2 + 5 = 5
Hence, the focus of the parabolic mirror is located at (x, y) = (3, 5).
In conclusion, any beam of light hitting the mirror from the y^- direction will be reflected toward the focus of the parabolic mirror, which is (3, 5).