If two lines $a 1 x + b_{1} y + C_{1} = 0$ & $t a_{2} x + b_{2} y + c_{2} = 0$ the co-ordinate axis in cyclic point prove that $a_{1} a_{2} = b_{1} b_{2}$ & Find the equation of circle?

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Solution

$2$ lines $a_{1} x + b_{1} y + c_{1} = 0$
& $a_{2} x + b_{2} y + c_{2} = 0$
$p (- \frac{c_{1}}{a_{1}}, 0), Q (0, \frac{- c_{1}}{b_{1}})$
$∴ a_{1} x + 0 + c_{1} = 0$
$R (\frac{- c_{2}}{a_{2}}, 0)$ , $s (0, \frac{- c_{2}}{b_{2}})$
$x = \frac{- c_{1}}{a_{1}}$
$\Rightarrow O R . O P = O Q . O S$
$∣ ∣ ∣ (\frac{- c_{1}}{a_{1}}) (\frac{- c_{2}}{a_{2}}) ∣ ∣ ∣ = ∣ ∣ ∣ (\frac{- c_{1}}{a_{1}}) (\frac{- c_{2}}{a_{2}}) ∣ ∣ ∣$
$\Rightarrow | a_{1} a_{2} | = | b_{1} b_{2} |$ .

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