Circles
Circle
Allen Star Batch
Grade 11
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 be:
$\left[0, \frac{1}{4}\right]$
$\left[0, \frac{2-\sqrt{2}}{4}\right]$
$\left[0, \frac{1}{2\sqrt{2}}\right]$
None of these
Step-by-Step Solution
Key Concept: The maximum common difference occurs when the smaller circle is internally tangent to the line.
For maximum common difference, the smaller circle just touches the line. Setting $1 - 2d = \frac{1}{\sqrt{2}}$ gives $d = \frac{\sqrt{2}-1}{2\sqrt{2}}$. Therefore, $d \in \left(0, \frac{\sqrt{2}-1}{2\sqrt{2}}\right)$, which simplifies to the interval $\left(0, \frac{2-\sqrt{2}}{4}\right)$.
Correct Answer: 2