Three concentric circles of which biggest is ${\mathrm{x}}^2+{\mathrm{y}}^2=1,$ have their radii in A.P. If the line $\mathrm{y}=\mathrm{x}+1$ cuts all the circles in real and distinct points, then the interval in which the common difference of A.P will lie, is

$\text{ Clearly centre at }\mathrm{O}(0,0)\text{ Let the radii of other two circles be }1-r,1-2r\text{, where }r>0\text{. }$

$$\begin{gathered} \text{ The circles are } \\ \begin{array}{ll}{x}^2+y^2=1& ----\left(1\right)\\ x^2+y^2=(1-r{)}^2& ----\left(2\right)\\ x^2+y^2=(1-2r{)}^2& ----\left(3\right)\end{array} \end{gathered}$$

$y=x+1\text{ cuts the circle(1) at }(1,0)\text{ and }(0,1)\text{. This will cuts circles (2) \&(3) in real and }$

$\text{ distinct points }$

$\left|\frac{1}{\sqrt{2}}\right|<1-r\text{ and }\left|\frac{1}{\sqrt{2}}\right|<1-2r$

$\Rightarrow r<1-\frac{1}{\sqrt{2}}\text{ and }r<\frac{1}{2}\left(1-\frac{1}{\sqrt{2}}\right)$

$\Rightarrow r\in \left(0,\frac{1}{2}\left(1-\frac{1}{\sqrt{2}}\right)\right)$