Question

A parallel beam of light ray parallel to the x-axis is incident on a parabolic reflecting surface $x=2by^{2}$ as shown in the figure. After reflecting it passes through focal point F. What is the focal length of the reflecting surface?

• A. $\frac{1}{2b}$
• B. $\frac{1}{8b}$
• C. $\frac{1}{ab}$
• D. $\frac{1}{b}$

Hint:

Always rearrange the given equation into the standard form to identify geometric parameters.

Answer 1

The Correct Option is B

Step 1: Concept
The standard equation of a parabola is $y^{2} = 4ax$, where $a$ is the focal length.
Step 2: Analysis
Given equation: $x = 2by^{2}$, which can be rewritten as $y^{2} = \frac{1}{2b}x$.
Step 3: Calculation
Comparing with $y^{2} = 4ax$: $4a = \frac{1}{2b}$. Solving for $a$: $a = \frac{1}{8b}$.
Step 4: Conclusion
Hence, the focal length is $\frac{1}{8b}$.
Final Answer:(B)