<p>If one of the diameters of the circle, given by the equation \(x^2 + y^2 - 4x + 6y - 12 = 0\), is a chord of a circle S, whose centre is at \((-3, 2)\), then the radius of S is</p>
Step-by-Step Solution
Key Concept: The diameter of the given circle becomes a chord of circle S. The perpendicular distance from center of S to this chord, combined with half the chord length, gives the radius using the chord-distance relationship: r² = d² + (chord/2)².
<p><strong>Step 1:</strong> Rewrite the given circle equation in standard form:</p><p>x² + y² - 4x + 6y - 12 = 0</p><p>(x² - 4x + 4) + (y² + 6y + 9) - 12 - 4 - 9 = 0</p><p>(x - 2)² + (y + 3)² = 25</p><p>Center: C₁ = (2, -3), Radius R₁ = 5</p><p><strong>Step 2:</strong> The diameter of this circle is a chord of circle S. Any diameter has length 2R₁ = 10.</p><p>The diameter lies on a chord line. Find the equation of a diameter. All diameters pass through (2, -3).</p><p><strong>Step 3:</strong> For the perpendicular distance from S's center (-3, 2) to any diameter of the first circle:</p><p>Distance from (-3, 2) to (2, -3) = √[(−3−2)² + (2−(−3))²] = √[25 + 25] = √50 = 5√2</p><p><strong>Step 4:</strong> The chord length is 10 (the diameter). Using the relationship for a chord:</p><p>If d = perpendicular distance from center S to the chord, and chord length = 2c, then:</p><p>r² = d² + c²</p><p>The minimum distance from (-3, 2) to any point on the diameter = 5√2 (when perpendicular).</p><p>With half-chord = 5: r² = (5√2)² - 5² is incorrect approach.</p><p><strong>Step 4 (Corrected):</strong> Using r² = d² + (chord/2)²:</p><p>r² = (5√2)² + 5² = 50 + 25 = 75</p><p>r = √75 = 5√3</p><p>∴ Answer: C (5√3)</p>
Correct Answer: C