Straight Line Question 336

Question: If the equation $ y=mx+c $ and $ x\cos \alpha +y\sin \alpha =p $ represents the same straight line, then

Options:

A) $ p=c\sqrt{1+m^{2}} $

B) $ c=p\sqrt{1+m^{2}} $

C) $ cp=\sqrt{1+m^{2}} $

D) $ p^{2}+c^{2}+m^{2}=1 $

Show Answer

Answer:

Correct Answer: B

Solution:

  • If the given lines represent the same line, then the length of the perpendiculars from the origin to the lines are equal, so that $ \frac{c}{\sqrt{1+m^{2}}}= $ $ \frac{p}{\sqrt{{{\cos }^{2}}\alpha +{{\sin }^{2}}\alpha }} $

$ \Rightarrow $ $ c=p\sqrt{1+m^{2}} $ .