<p>A circle passes through the points \((2, 3)\) and \((4, 5)\). If the centre lies on the line \(y - 4x + 3 = 0\), then its radius is equal to</p>
Step-by-Step Solution
Key Concept: The center of the circle is equidistant from all points on the circle. Use the condition that the center lies on the given line, then apply the distance formula from the center to any point on the circle to find the radius.
<p><strong>Step 1:</strong> Let the center be C(h, k). Since it lies on y - 4x + 3 = 0, we have:</p><p>k = 4h - 3 ... (i)</p><p><strong>Step 2:</strong> Since the circle passes through (2, 3) and (4, 5), the center is equidistant from both points:</p><p>(h - 2)² + (k - 3)² = (h - 4)² + (k - 5)²</p><p><strong>Step 3:</strong> Expand both sides:</p><p>h² - 4h + 4 + k² - 6k + 9 = h² - 8h + 16 + k² - 10k + 25</p><p><strong>Step 4:</strong> Simplify:</p><p>-4h - 6k + 13 = -8h - 10k + 41</p><p>4h + 4k = 28</p><p>h + k = 7 ... (ii)</p><p><strong>Step 5:</strong> Substitute (i) into (ii):</p><p>h + 4h - 3 = 7</p><p>5h = 10</p><p>h = 2, k = 5</p><p><strong>Step 6:</strong> Center is C(2, 5). Find radius using distance to (2, 3):</p><p>r = √[(2-2)² + (5-3)²] = √4 = 2</p><p>∴ Answer: C</p>
Correct Answer: C