<p>If a circle <em>C</em>, whose radius is 3, touches externally the circle, <em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> + 2<em>x</em> − 4<em>y</em> − 4 = 0 at the point (2, 2), then the length of the intercept cut by this circle <em>C</em>, on the <em>x</em>-axis is equal to</p>
Step-by-Step Solution
Key Concept: For two externally tangent circles, the line joining their centers passes through the point of tangency. Use this to find the center of circle C, then apply the chord-intercept formula.
<p><strong>Step 1:</strong> Rewrite the given circle in standard form: x² + y² + 2x − 4y − 4 = 0</p><p>(x+1)² + (y−2)² = 9</p><p>Center O₁ = (−1, 2), radius r₁ = 3</p><p><strong>Step 2:</strong> For external tangency at point P(2, 2), the center C of circle C lies on the line O₁P extended beyond P. The direction vector is (2−(−1), 2−2) = (3, 0), so the line is horizontal: y = 2</p><p><strong>Step 3:</strong> Distance |O₁P| = √[(2+1)² + (2−2)²] = 3</p><p>For external tangency: |O₁C| = r₁ + r₂ = 3 + 3 = 6</p><p>Since C lies on the ray from O₁ through P at distance 6 from O₁, and the direction is (3, 0), we have C = (−1 + 6, 2) = (5, 2)</p><p><strong>Step 4:</strong> Circle C has center (5, 2) and radius 3: (x−5)² + (y−2)² = 9</p><p><strong>Step 5:</strong> For x-axis intercept, set y = 0:</p><p>(x−5)² + (0−2)² = 9</p><p>(x−5)² + 4 = 9</p><p>(x−5)² = 5</p><p>x = 5 ± √5</p><p><strong>Step 6:</strong> Length of intercept = (5 + √5) − (5 − √5) = 2√5</p><p>∴ Answer: A (2√5)</p>
Correct Answer: A