Applications Of Derivatives Question 7
Question: If $ xcos\theta +ysin\theta =2 $ is perpendicular to the line $ x-y=3 $ , then what is one of the value of $ \theta $ -
Options:
A) $ \pi /6 $
B) $ \pi /4 $
C) $ \pi /2 $
D) $ \pi /3 $
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Answer:
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 =2is-\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} $
BETA
