Question

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

• A. $\left(0, \frac{1}{4}\right)$
• B. $\left(0, \frac{1}{2\sqrt{2}}\right)$
• C. $\left(0, \frac{2-\sqrt{2}}{4}\right)$
• D. none

Answer 1

Answer: (C)

$\left(0, \frac{2-\sqrt{2}}{4}\right)$

Answer 2

Let the radii of the three concentric circles be $r_1, r_2, r_3$. The circles are concentric, so their center is at the origin (0,0). The largest circle is $x^2 + y^2 = 1$, so its radius is $R_{max} = 1$. The radii are in an Arithmetic Progression (A.P.). Let the common difference be $d$. Since 1 is the largest radius, the radii must be in decreasing order. Let the radii be $1, 1-d, 1-2d$. For these to be valid positive radii, we must have $1-2d > 0$, which implies $d < 1/2$. For distinct radii, $d \ne 0$. Since 1 is the largest radius, $d$ must be positive. Thus, $0 < d < 1/2$.

The equation of the line is $y = x+1$, which can be written as $x - y + 1 = 0$. The distance $p$ from the center of the circles (0,0) to the line is given by: $p = \frac{|1(0) - 1(0) + 1|}{\sqrt{1^2 + (-1)^2}} = \frac{1}{\sqrt{2}}$

For the line to cut all three circles in real and distinct points, the distance $p$ must be less than each radius: $p < r_i$ for $i=1, 2, 3$. This implies $p < \min(r_i)$. The smallest radius is $1-2d$. So, we must have: $\frac{1}{\sqrt{2}} < 1-2d$

Rearranging this inequality to solve for $d$: $2d < 1 - \frac{1}{\sqrt{2}}$ $d < \frac{1}{2}\left(1 - \frac{1}{\sqrt{2}}\right)$

We also have the condition $0 < d < 1/2$. Let's evaluate the upper bound: $\frac{1}{2}\left(1 - \frac{1}{\sqrt{2}}\right) = \frac{1}{2}\left(\frac{\sqrt{2}-1}{\sqrt{2}}\right) = \frac{\sqrt{2}-1}{2\sqrt{2}}$ To rationalize the denominator, multiply by $\frac{\sqrt{2}}{\sqrt{2}}$: $\frac{(\sqrt{2}-1)\sqrt{2}}{2\sqrt{2}\sqrt{2}} = \frac{2-\sqrt{2}}{4}$

Since $\frac{2-\sqrt{2}}{4} < \frac{1}{2}$, the condition $d < 1/2$ is automatically satisfied. Therefore, the interval for the common difference $d$ is $0 < d < \frac{2-\sqrt{2}}{4}$.