Question

Three concentric circles of which the biggest is $x^2+y^2=1$ have their radii in A.P. with common difference $d(>0)$. If the line $y=x+1$ cuts all the circles in real distinct points, then

• A. $d<\frac{2-\sqrt{2}}{4}$
• B. $d>\frac{2+\sqrt{2}}{4}$
• C. $d>1+\frac{1}{\sqrt{2}}$
• D. $d$ is any real number

Answer 1

The correct option is A $d<\frac{2-\sqrt{2}}{4}$

Let the radius of the circle are $r_1,r_2$ and 1
line $y=x+1$
Perpendicular from $(0,0)$ on line $y=x+1$ $=\frac{1}{\sqrt{2}}$
Now $r_1>\frac{1}{\sqrt{2}}$ but $r_1=1-2d$
hence $1-2d>\frac{1}{\sqrt{2}}\Rightarrow \frac{\sqrt{2}-1}{\sqrt{2}}>2d\Rightarrow d<\frac{\sqrt{2}-1}{2\sqrt{2}}$