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If two lines \mathrm{a_1 x+b_1 y+c_1=0} and \mathrm{a_2 x+b_2 y+c_2=0} cut the coordinate axes at concyclic points, then
 

Option: 1

\mathrm{a_1 a_2+b_1 b_2=0}

 


Option: 2

\mathrm{a_1 a_2-b_1 b_2=0}


Option: 3

\mathrm{a_1 b_1+a_2 b_2=0}


Option: 4

\mathrm{a_1 b_1-a_2 b_2=0}


Answers (1)

best_answer

The given lines meet x-axis at \mathrm{\left(-c_1 / a_1, 0\right)} and \mathrm{\left(-c_2 / a_2, 0\right)}, y-axis at \mathrm{\left(0,-c_1 / b_1\right)} and \mathrm{\left(0,-c_2 / b_2\right).}
If the equation of the circle passing through these points is \mathrm{x^2+y^2+2 g x+2 f y+c=0, }then \mathrm{-c_1 / a_1,-c_2 / a_2} are the roots of\mathrm{x^2+2 g x+c=0 }

\mathrm{\Rightarrow c=c_1 c_2 / a_1 a_2.}
Similarly \mathrm{c=c_1 c_2 / b_1 b_2 \Rightarrow a_1 a_2-b_1 b_2=0.}

Hence option 2 is correct.

Posted by

Shailly goel

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