Question

Coordinates of the centre of a circle whose radius is $2$ unit 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. $(0,2\sqrt{2})$

Answer 1

Answer: (B)

$(1+2\sqrt{2},0)$

Let center be $(a,b)$

Given $r=2$

The pair of lines can be written as

$(-x+y+1)(-x-y+1)=0$

$⟹x=y+1,x+y=1$ are tangents

Radius = perpendicular distance from center to the tangents

$2=\frac{|a–b-1|}{\sqrt{1^2+1^2}}$

$⟹a-b=2\sqrt{2}+1-eq.1$

And

$2=\frac{|a+b-1|}{\sqrt{1^2+1^2}}$

$⟹a+b=2\sqrt{2}+1-eq.2$

From eq.1 and eq.2

$a=2\sqrt{2}+1,b=0$

$\text{Center}=(2\sqrt{2}+1,0)$