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

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

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

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

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Solution

The correct option is A $a_{1} a_{2} = b_{1} b_{2}$
The line $a_{1} x + b_{1} y + c_{1} = 0$ cuts the coordinate axes at $A (- \frac{c_{1}}{a_{1}}, 0)$ and $B (0, - \frac{c_{1}}{b_{1}})$ .

And the line $a_{2} x + b_{2} y + c_{2} = 0$ cuts the axes at $C (- \frac{c_{2}}{a_{2}}, 0)$ and $D (0, - \frac{c_{2}}{b_{2}})$ .

So $A C$ and $B D$ are chords along $x -$ axis and $y -$ axis, respectively, intersecting at origin $O$ .

Since $A, B, C, D$ are concyclic, therefore $O A \cdot O C = O B \cdot O D$

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

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

Hence, option A.

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