<p>The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2, 3) is</p>
Step-by-Step Solution
Key Concept: A circle touching the x-axis at (1,0) has its center on the vertical line x=1. The radius equals the y-coordinate of the center, so use the condition that the circle passes through (2,3) to find the center and radius.
<p><strong>Step 1:</strong> If the circle touches the x-axis at (1, 0), the center must be at (1, r) where r is the radius (since the circle is tangent to the x-axis).</p><p><strong>Step 2:</strong> The circle passes through point (2, 3), so the distance from center (1, r) to (2, 3) equals the radius r:</p><p>√[(2-1)² + (3-r)²] = r</p><p><strong>Step 3:</strong> Squaring both sides:</p><p>1 + (3-r)² = r²</p><p>1 + 9 - 6r + r² = r²</p><p>10 - 6r = 0</p><p>r = 5/3</p><p><strong>Step 4:</strong> The diameter = 2r = 2(5/3) = <strong>10/3</strong></p><p>∴ Answer: A</p>
Correct Answer: A