(Solution) - The diagram at right shows the reflection of a ray -(2025 Original AI-Free Solution)

Paper Details

The diagram at right shows the reflection of a ray of light in a parabolic reflector. The angles A and B are equal. Follow these steps to verify this property of parabolas.
a. Sketch the parabola y2 = 8x.

b. What are the coordinates of the focus of this parabola?
c. On the same graph, sketch the line y = 2x + 1, tangent to the parabola. Find the coordinates of the point of tangency. What is the slope of this line?
d. Sketch a ray through the point of tangency parallel to the axis of symmetry. What is the slope of this line?
e. Draw the segment from the point of intersection to the focus. What is the slope of this segment?
f. The formula tan A = m2 - m1 / 1 +m2m1 applies when A is the angle between two lines with slopes m1 and m2. Use this formula to find the angle between the tangent line and the horizontal line. Then find the angle between the tangent line and the segment joining the focus and the point of tangency. What do you notice about the angles?