Question

If the straight line $x\cos \theta +y\sin \theta =2$ touches the circle $x^2+y^2-2x=0$ then

• A. $\theta =n\pi ,n\in Z$
• B. $\theta =(2n+1)\pi ,n\in Z$
• C. $\theta =2n\pi ,n\in Z$
• D. $\theta =(2n+1)\frac{\pi }{2},n\in Z$

Answer 1

Answer: (C)

$\theta =2n\pi ,n\in Z$

$x^2+y^2-2x=0$
Centre $\equiv (1,0)$ and radius $=1$

If the line touches the circle, then
$\left|\frac{1\cos \theta -2}{\sqrt{{\sin }^2\theta +{\cos }^2\theta }}\right|=1$

$\Rightarrow \cos \theta =3$ or $\cos \theta =1$
$\Rightarrow \theta =2n\pi ,n\in Z$