The line $(x-2)\cos \theta +(y-2)\sin \theta =1$ touches a circle for all value of $\theta$, then the equation of circle is

Given line is

$(x-2)cos \,\theta +(y-2)sin\, \theta =1$

$\Rightarrow (x-2)cos \theta +(y-2)sin\theta=cos^2 \, \theta+sin^2\, \theta$

On compaining we get,

$(x-2) = cos\theta$ ..... $(i)$

$(y-2) = sin \theta$ ..... $(ii)$

On squaring and then adding Eqs. $(i)$ and $(ii),$ we get

$(x-2)^2+(y-2)^2=cos^2\theta +sin^2 \theta$

$\Rightarrow (x-2)^2+(y-2)^2=1$

$\Rightarrow x^2+y^2-4x-4y+7=0$