<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: When a circle touches the x-axis at a point, the center lies on the vertical line through that point. Use this constraint to find the center and radius.
<p>Since the circle touches the x-axis at \((3, 0)\), its center lies on the line \(x = 3\). Let the center be \((3, r)\) where \(r\) is the radius. The circle passes through \((1, -2)\), so \((1-3)^2 + (-2-r)^2 = r^2\). Solving: \(4 + 4 + 4r + r^2 = r^2\), giving \(r = -2\). The circle equation is \((x-3)^2 + (y+2)^2 = 4\). Testing \((5, -2)\): \((5-3)^2 + (-2+2)^2 = 4 + 0 = 4\). ✓</p><p>∴ Answer is (d).</p>
Correct Answer: d