If the lines $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ cut the coordinates axes in concyclic points, then

$a_{1} b_{1} = a_{2} b_{2}$

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$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}$

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$a_{1} + a_{2} = b_{1} + b_{2}$

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$a_{1} a_{2} = b_{1} b_{2}$

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Solution

The correct option is D
$a_{1} a_{2} = b_{1} b_{2}$

Let the given lines be $L_{1} \equiv a_{1} x + b_{1} y + c_{1} = 0$ and $L_{2} \equiv a_{2} x + b_{2} y + c_{2} = 0$ . Suppose $L_{1}$ meets the coordinate axes at A and B and $L_{2}$ meets at C and D. Then coordinates of A, B, C, D are
$A (- \frac{c_{1}}{a_{1}}, 0), B (0, - \frac{c_{1}}{b_{1}}) C (- \frac{c_{2}}{a_{2}}, 0), and D (0, - \frac{c_{2}}{b_{2}})$
Since A, B, C, D are concyclic, therefore OA.OC=OD.OB

$\Rightarrow (- \frac{c_{1}}{a_{1}}) (- \frac{c_{2}}{a_{2}}) = (\frac{c_{2}}{b_{2}}) (- \frac{c_{1}}{b_{1}}) \Rightarrow a_{1} a_{2} = b_{1} b_{2}$

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