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  • If the lines <img alt="\mathrm{a} 1 \mathrm{x}+\mathrm{b} 1 \mathrm{y}+\mathrm{c} 1=0" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Ba%7D%201%20%5Cmathrm%7Bx%7D&plus;%5Cmathr

If the lines \mathrm{a} 1 \mathrm{x}+\mathrm{b} 1 \mathrm{y}+\mathrm{c} 1=0 and \mathrm{a} 2 \mathrm{x}+\mathrm{b} 2 \mathrm{y}+\mathrm{c} 2=0 cut the coordinate axes in concyclic points, then

Option: 1

\mathrm{a} 1 \mathrm{a} 2=\mathrm{b} 1 \mathrm{~b} 2


Option: 2

\mathrm{a} 1 \mathrm{~b} 1=\mathrm{a} 2 \mathrm{~b} 2


Option: 3

\mathrm{a1b2 = a 2 \mathrm{~b} 1}


Option: 4

None of these


Answers (1)

best_answer

Let the given lines be L1 \mathrm{a} 1 \mathrm{x}+\mathrm{b} 1 \mathrm{y}+\mathrm{c} 1=0 and \mathrm{L} 2\: \mathrm{a} 2 \mathrm{x}+\mathrm{b} 2 \mathrm{y}+\mathrm{c} 2=0.Suppose L1 meets the coordinate axes at \mathrm{P}and \mathrm{Q} and \mathrm{L} 2 meets at \mathrm{R} and \mathrm{S}. Then the coordinates of \mathrm{P}, \mathrm{Q}, \mathrm{S} are

\mathrm{P\left(-\frac{c_{1}}{a_{1}}, 0\right), Q\left(0, \frac{-c_{1}}{b_{1}}\right), R\left(\frac{-c_{2}}{a_{2}}, 0\right), S\left(0,-\frac{c_{2}}{b_{2}}\right)}

Since \mathrm{P, Q, R, S} are concyclic, therefore,
\mathrm{\mathrm{OP} . \mathrm{OR}=\mathrm{OQ} \cdot \mathrm{OS}}

\mathrm{\Rightarrow\left(-\frac{c_{1}}{a_{1}}\right)\left(-\frac{c_{2}}{a_{2}}\right)=\left(-\frac{c_{1}}{b_{1}}\right)\left(-\frac{c_{2}}{b_{2}}\right)}
\mathrm{\Rightarrow \mathrm{a} 1 \mathrm{a} 2=-\mathrm{b} 1 \mathrm{~b} 2}.

Posted by

Divya Prakash Singh

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