Question

The straight line $x\cos \theta +y\sin \theta =2$, will touch the circle $x^2+y^2-2x=0$, if

• A. $\theta =n\pi ,n\in Z$
• B. $\theta =(2n+1)\pi ,n\in Z$
• C. $\theta =2n\pi ,n\in Z$
• D. None of these

Answer 1

The correct option is C $\theta =2n\pi ,n\in Z$

$x^2+y^2-2x=0$
$C=(1,0),r=\sqrt{1^2}=1$
Now,
The line $x\cos \theta +y\sin \theta =2$ is tangent on the circle if distance from centre to line is equal to radius, so
$\left|\frac{\cos \theta -2}{\sqrt{{\cos }^2\theta +{\sin }^2\theta }}\right|=1\Rightarrow \cos \theta -2=\pm 1\Rightarrow \cos \theta =1,3\therefore \cos \theta =1(\because \cos \theta \in [-1,1\left]\right)\Rightarrow \theta =2n\pi ,n\in Z$