Question Detail

First find the center of the given circle: x² + y² - 3x - 4y - 1 = 0 → (x - 3/2)² + (y - 2)² = (9/4) + 4 + 1 = (9/4) + 5 = (9+20)/4 = 29/4. Center C₁ = (3/2, 2). The new circle has center C₂ = (5, 2) and passes through C₁. Radius = distance between centers = |5 - 3/2| = |(10-3)/2| = 7/2 = 3.5. Equation: (x-5)² + (y-2)² = (7/2)² = 49/4 → x² - 10x + 25 + y² - 4y + 4 = 49/4 → x² + y² - 10x - 4y + 29 = 49/4 → multiply by 4: 4x² + 4y² - 40x - 16y + 116 = 49 → 4x² + 4y² - 40x - 16y + 67 = 0. Divide by 4: x² + y² - 10x - 4y + 16.75 = 0, which is not among the options. Option B: x² + y² - 10x - 4y + 13 = 0 would give center (5,2) and radius² = 25+4-13 = 16, radius = 4. That circle would pass through (5,2±4) or (5±4,2), not through (1.5,2).