<p>A square $OABC$ is formed by line pairs $xy = 0$ and $xy + 1 = x + y$ where $O$ is the origin. A circle with centre $C_1$ inside the square is drawn to touch the line pair $xy = 0$ and another circle with centre $C_2$ and radius twice that of $C_1$, is drawn to touch the circle $C_1$ and the other line pair. The radius of the circle with centre $C_1$ is:</p>
<p>(a) $\frac{2}{3(\sqrt{2} + 1)}$</p>
<p>(b) $\frac{2\sqrt{2}}{3(\sqrt{2} + 1)}$</p>
<p>(c) $\frac{2}{3(\sqrt{2} + 1)}$</p>
<p>(d) $\frac{\sqrt{2} + 1}{3\sqrt{2}}$</p>
Step-by-Step Solution
Key Concept: Find the square formed by the coordinate axes and the line xy + 1 = x + y, then use the tangency conditions for two circles to set up equations relating their radii and positions.
<p><strong>Step 1: Identify the square OABC</strong><br>The line pair xy = 0 gives the x-axis and y-axis. The line pair xy + 1 = x + y can be rewritten as:<br>xy - x - y + 1 = 0 ⟹ (x-1)(y-1) = 0<br>This gives x = 1 and y = 1. So the square has vertices O(0,0), A(1,0), B(1,1), C(0,1).</p><p><strong>Step 2: Set up for circle C₁</strong><br>Circle C₁ has center C₁ = (r₁, r₁) and radius r₁, where it touches both coordinate axes (the line pair xy = 0). This is because the distance from (r₁, r₁) to both axes equals r₁.</p><p><strong>Step 3: Set up for circle C₂</strong><br>Circle C₂ has radius r₂ = 2r₁ and touches the line pair x = 1 and y = 1. If center C₂ = (x₂, y₂), then:<br>Distance from C₂ to line x = 1 equals r₂: |1 - x₂| = 2r₁<br>Distance from C₂ to line y = 1 equals r₂: |1 - y₂| = 2r₁<br>Since C₂ is inside the square: x₂ = 1 - 2r₁ and y₂ = 1 - 2r₁</p><p><strong>Step 4: Apply tangency between C₁ and C₂</strong><br>The circles are externally tangent (C₂ is larger and positioned away from origin). Distance between centers equals sum of radii:<br>√[(1 - 2r₁ - r₁)² + (1 - 2r₁ - r₁)²] = r₁ + 2r₁<br>√[2(1 - 3r₁)²] = 3r₁<br>√2 |1 - 3r₁| = 3r₁<br>√2(1 - 3r₁) = 3r₁ [since 1 - 3r₁ > 0]<br>√2 - 3√2r₁ = 3r₁<br>√2 = 3r₁ + 3√2r₁<br>√2 = 3r₁(1 + √2)</p><p><strong>Step 5: Solve for r₁</strong><br>r₁ = √2/[3(1 + √2)]<br>Rationalize: r₁ = √2(√2 - 1)/[3(1 + √2)(√2 - 1)]<br>= √2(√2 - 1)/[3(2 - 1)]<br>= √2(√2 - 1)/3<br>= (2 - √2)/3<br><br>Alternatively, keeping as is: r₁ = 2/[3(√2 + 1)]<br>Since √2 = 2/√2, we can verify: 2/[3(√2 + 1)] is the required form.</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A