A ray of light is coming along the line $y = b$ from the positive direction of $x -$ axis and strikes the concave mirror whose interaction with $x -$ plane is a parabola $y^{2} = 4 a x$ . Find the equation of the reflected ray. Assume $a, b$ are positive.

(y - b) (4 a^{2} - b^{2}) = (4 a x + b^{2})

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(y + b) (4 a^{2} - b^{2}) = (4 a x - b^{2})

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(y + b) (4 a^{2} + b^{2}) = - (4 a x - b^{2})

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(y - b) (4 a^{2} - b^{2}) = - (4 a x - b^{2})

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Solution

The correct option is D $(y - b) (4 a^{2} - b^{2}) = - (4 a x - b^{2})$
Reflected ray passes through focus of parabola $(a, 0)$ and $(\frac{b^{2}}{4 a}, b)$
$y = b, x = \frac{b^{2}}{4 a}$
$\Rightarrow \frac{y - 0}{x - a} = \frac{b - 0}{\frac{b^{2}}{4 a} - a}$
$\Rightarrow \frac{y - b}{x - \frac{b^{2}}{4 a}} = \frac{b - 0}{\frac{b^{2}}{4 a} - a}$
$\Rightarrow \frac{y - b}{(4 a x - b^{2})} = \frac{b}{(b^{2} - 4 a^{2})}$

$\Rightarrow (y - b) (4 a^{2} - b^{2}) = - b (4 a x - b^{2})$

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