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If the straight line \mathrm{a x+b y+c=0} and \mathrm{x \cos \alpha+y \sin \alpha=c}, enclose an angle \mathrm{\frac{\pi}{4}} between them and meet the straight line \mathrm{x \sin \alpha-y \cos \alpha=0} in the same point, then

Option: 1

\mathrm{a^{2}+b^{2}=c^{2}}


Option: 2

\mathrm{a^{2}+b^{2}=2}


Option: 3

\mathrm{a^{2}+b^{2}=2 c^{2}}


Option: 4

\mathrm{a^{2}+b^{2}=4}


Answers (1)

best_answer

\mathrm{O M=c} (Clear from normal form of the line)
\mathrm{O N=\frac{c}{\sqrt{a^{2}+b^{2}}}}

\mathrm{Also \: \angle O M N=45^{\circ}}

\mathrm{So,\: O N=O M \cos 45^{\circ}}
\mathrm{\begin{gathered} \frac{c}{\sqrt{a^{2}+b^{2}}}=\frac{c}{\sqrt{2}} \\ \Rightarrow \quad a^{2}+b^{2}=2 \end{gathered}}

Posted by

avinash.dongre

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