<p>The tangent to the circle \(C_1 : x^2 + y^2 - 2x - 1 = 0\) at the point \((2, 1)\) cuts off a chord of length 4 from a circle \(C_2\) whose centre is \((3, -2)\). The radius of \(C_2\) is</p>
Step-by-Step Solution
Key Concept: Use the perpendicular distance from the center of C₂ to the tangent line of C₁, then apply the chord-length formula: if distance from center to chord is d and chord length is 2l, then r² = d² + l².
<p><strong>Step 1:</strong> Rewrite C₁ in standard form: (x−1)² + y² = 2, so center O₁ = (1, 0) and radius = √2.</p><p><strong>Step 2:</strong> Find the tangent to C₁ at (2, 1). The radius O₁(2,1) has slope (1−0)/(2−1) = 1, so the tangent (perpendicular to radius) has slope −1.</p><p>Tangent equation: y − 1 = −1(x − 2) → <strong>x + y − 3 = 0</strong></p><p><strong>Step 3:</strong> Find perpendicular distance from C₂'s center (3, −2) to the tangent line x + y − 3 = 0:</p><p>d = |3 + (−2) − 3|/√(1² + 1²) = |−2|/√2 = 2/√2 = √2</p><p><strong>Step 4:</strong> Use the chord-length formula. If chord length = 4, then half-chord length l = 2. For a circle with radius r and perpendicular distance d from center to chord:</p><p>r² = d² + l² = (√2)² + 2² = 2 + 4 = 6</p><p>∴ r = √6</p><p><strong>Answer: D</strong></p>
Correct Answer: D