Question

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

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

Answer 1

The correct option is A $(0,\frac{\sqrt{2}-1}{2\sqrt{2}})$

Let the common difference of the radii are $r,>0$
then the radius of other two circles will be $1-r,1-2r$
Now if the line $y=x+1$ intersects the smallest radius $(1-2r)$ circle, we can conclude that it will surely intersect the other two circles also
Now for the condition to satisfy
$1-2r>\frac{1}{\sqrt{2}},1-2r>0\Rightarrow r\in (0,\frac{\sqrt{2}-1}{2\sqrt{2}})$