Question

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

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

Answer 1

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

$x^2+y^2-2x=0$
$\Rightarrow (x-1{)}^2+y^2=1$
Thus, centre of circle is $(1,0)$ and radius is $1$.
The given line will touch the circle, only if the distance from centre of the circle to the line is equal to the radius.
Thus, $|\cos \theta -2|=1\Rightarrow cos\theta =1,3$.
Thus, $\cos \theta =1\Rightarrow \theta =2n\pi ,n\in I$
Hence, option 'C' is correct.