A circle passes through the origin $0$ and cuts the axis at $A (a . 0)$ and $B (0, b)$ . The reflection of $O$ in the line $A B$ is the point

(\frac{2 a b^{2}}{a^{2} + b^{2}}, \frac{2 a^{2} b}{a^{2} + b^{2}})

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(\frac{2 a^{2} b}{a^{2} + b^{2}}, \frac{2 a b^{2}}{a^{2} + b^{2}})

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(\frac{2 a b}{a^{2} + b^{2}}, \frac{2 a b}{a^{2} + b^{2}})

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(a, b)

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Solution

The correct option is D $(\frac{2 a b^{2}}{a^{2} + b^{2}}, \frac{2 a^{2} b}{a^{2} + b^{2}})$
Equation of $A B$ is $\frac{x}{a} + \frac{y}{b} = 1$
If $P (h, k)$ is the reflextion of $O$ in $A B$ then $(\frac{h}{2}, \frac{k}{2})$
lies on $A B$ and $A B$ and $O P$ are at right angles.
$∴ \frac{h}{a} + \frac{k}{b} = 2$ and $- \frac{b}{a} \times \frac{k}{h} = - 1$
$\Rightarrow h = \frac{2 a b^{2}}{a^{2} + b^{2}}$ and $k = \frac{2 a^{2} b}{a^{2} + b^{2}}$

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