Question

The coordinates of the centre of a circle, whose radius is $2$ units and which touches the line pair $x^2-y^2-2x+1=0,$ are

• A. $(4,0)$
• B. $(1+2\sqrt{2},0)$
• C. $(4,1)$
• D. $(1,2\sqrt{2})$

Answer 1

Answer: (B), (D)

The correct options are
B $(1+2\sqrt{2},0)$
D $(1,2\sqrt{2})$

Line pair is $(x-1{)}^2-y^2=0,$
i.e., $x+y-1=0,x-y-1=0$


If the centre is on the $x$-axis,
let the centre be $(\alpha ,0).$
Then its distance from $x+y-1=0$ is
$\left|\frac{\alpha -1}{\sqrt{2}}\right|=2\text{(radius)}$
i.e., $\alpha =1\pm 2\sqrt{2}$
$\therefore$ Centre may be $(1+2\sqrt{2},0),(1-2\sqrt{2},0)$

If the centre is on the line $x=1,$
let the centre be $(1,\beta ).$
Then $\left|\frac{1+\beta -1}{\sqrt{2}}\right|=2$
$\Rightarrow \beta =\pm 2\sqrt{2}$
$\therefore$ Centre may be $(1,2\sqrt{2}),(1,-2\sqrt{2})$