Question

If the lines $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ cut the coordiante axes at concyclic points, then prove that $|a_1{a}_2|=b_1{b}_2|$.

Answer 1

$a_{1}x+b_{1}y+c_1=0$

cut coordinate axes at :

$(\frac{-c_1}{a_1},0)$ and $(0,\frac{-c_1}{b_1})$

$a_{2}x+b_{2}y+c_2=0$

cut coordinate axes at :

$(\frac{-c_2}{a_2},0)$ and $(0,\frac{-c_2}{b_2})$

$\therefore$ By property

$(\frac{-c_1}{a_1}\times \frac{-c_2}{a_2})=(\frac{-c_1}{b_1}\times \frac{c_2}{b_2})$

$=\left|\frac{c_1{c}_2}{a_1{a}_2}\right|=\left|\frac{c_1{c}_2}{b_1{b}_2}\right|$

$\therefore a_1{a}_2=b_1{b}_2$