A) \[\pi /6\]
B) \[\pi /4\]
C) \[\pi /2\]
D) \[\pi /3\]
Correct Answer: B
Solution :
| [b] Consider a line |
| \[x\,\,\cos \theta +y\,\sin \theta =2\] |
| \[\Rightarrow y\sin \theta =-x\cos \theta +2\] |
| \[\Rightarrow y=-x\frac{\cos \theta }{\sin \theta }+\frac{2}{\sin \theta }\] |
| \[\Rightarrow y=-x\cot \theta +2\cos ec\theta \] |
| On comparing this equation with |
| \[y=mx+c\] we get |
| slope of line \[x\,\,\cos \theta +y\,\,\sin \theta =2\,\,is\,\,-\cot \,\,\theta \] |
| Also, we have a line \[x-y=3\] is 1. |
| \[\Rightarrow \,\,\,\,\,\,\,\,\,y=x-3\] |
| slope of line \[x-y=3\] is 1. |
| Since, both the lines are perpendicular to each other |
| \[\therefore \] Product of their slopes = -1 |
| \[\Rightarrow (-cot\theta )(1)=-1\] |
| \[\Rightarrow cot\theta =1=\cot \frac{\pi }{4}\] |
| \[\Rightarrow \theta =\frac{\pi }{4}\] |
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