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If the straight lines \mathrm{a x+b y+P=0}and \mathrm{x \cos \alpha+y \sin \alpha=P}are inclined at an angle \mathrm{\pi / 4} and concurrent with straight line \mathrm{x \sin \alpha-y \cos \alpha=0}, then the value of \mathrm{a^{2}+b^{2}} is

Option: 1

1


Option: 2

0


Option: 3

2


Option: 4

13


Answers (1)

best_answer

\mathrm{ON}= distance of origin from the line \mathrm{x \sin \alpha +y\cos \alpha = p}
\mathrm{\mathrm{OM}=}Perpendicular distance of \mathrm{(0,0)} from the line \mathrm{a x+b y+P=0}

\mathrm{\Rightarrow \mathrm{OM}=-\frac{\mathrm{P}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}}
Now OMN is a right angle triangle with \mathrm{\angle \mathrm{ONM}=\pi / 4}

\mathrm{\Rightarrow O M=O N \sin \pi / 4=\frac{P}{\sqrt{2}}}
\mathrm{\Rightarrow a^{2}+b^{2}=2}
 

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