<p>The circle passing through (1, −2) and touching the axis of x at (3, 0) also passes through the point</p>
Step-by-Step Solution
Key Concept: A circle touching the x-axis at point (3,0) has its center directly above/below this point at (3,r) where r is the radius. Use the condition that (1,-2) lies on the circle to find r, then find other points on the circle.
<p><strong>Step 1:</strong> Since the circle touches the x-axis at (3, 0), the center lies on the perpendicular to the x-axis through (3, 0). Therefore, center is at C = (3, r) where r is the radius.</p><p><strong>Step 2:</strong> The circle passes through (1, -2), so distance from C to (1, -2) equals r:</p><p>√[(3-1)² + (r-(-2))²] = r</p><p>√[4 + (r+2)²] = r</p><p>4 + r² + 4r + 4 = r²</p><p>4r + 8 = 0</p><p>r = -2</p><p><strong>Step 3:</strong> Since r must be positive (radius), we take r = 2 (the center is at (3, -2) when the circle is below the x-axis).</p><p><strong>Step 4:</strong> Circle equation: (x-3)² + (y+2)² = 4</p><p>Expanding: x² - 6x + 9 + y² + 4y + 4 = 4</p><p>x² + y² - 6x + 4y + 9 = 0</p><p><strong>Step 5:</strong> Check given options by substituting into the circle equation. The point satisfying this equation is the answer.</p><p>∴ Answer: B</p>
Correct Answer: B