Question

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

• A. $x^2+y^2-4x-4y+7=0$
• B. $x^2+y^2+4x+4y+7=0$
• C. $x^2+y^2-4x-4y-7=0$
• D. None of the above

Answer 1

Answer: (A)

Give line is
$(x-2)\cos \theta +(y-2)\sin \theta =1={\cos }^2\theta +{\sin }^2\theta$ On
comparing we get,
$x-2\cos \theta \dots$..(i)
$y-2=\sin \theta \dots$...(ii)
and On squaring and then adding Eqs. (i) and (ii), we get
$(x-2{)}^2+(y-2{)}^2={\cos }^2\theta +{\sin }^2\theta$
$\Rightarrow (x-2{)}^2+(y-2{)}^2=1$
$\Rightarrow x^2+y^2-4x-4y+7=0$