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

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$\frac{1}{4 b}$

$\frac{1}{2 b}$

$\frac{1}{8 b}$

$\frac{1}{b}$

answer is B.

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Detailed Solution

given: $x = 2 b y^{2}$

general equation of a parabola: $y^{2} = 4 a x$ , where, (a,0) is focal point.

comparing these two equations, we get,

$4 a = \frac{1}{2 b}$

$\Rightarrow a = \frac{1}{8 b}$

hence, focal length: $f = \frac{1}{8 b}$

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