<p>If the centre \((\alpha, \beta)\) of a circle lies on the line \(y - 4x + 3 = 0\) and the circle passes through the points (2, 3) and (4, 5), find the radius of the circle.</p>
Step-by-Step Solution
Key Concept: The center lies on the given line, so β = 4α - 3. Since the circle passes through two points, these points are equidistant from the center—use this to find α, then calculate the radius.
<p><strong>Step 1:</strong> Since center (α, β) lies on y - 4x + 3 = 0, we have: β = 4α - 3</p><p><strong>Step 2:</strong> Both points (2, 3) and (4, 5) lie on the circle, so they're equidistant from center (α, β):</p><p>(2 - α)² + (3 - β)² = (4 - α)² + (5 - β)²</p><p><strong>Step 3:</strong> Expanding:</p><p>4 - 4α + α² + 9 - 6β + β² = 16 - 8α + α² + 25 - 10β + β²</p><p>13 - 4α - 6β = 41 - 8α - 10β</p><p>4α + 4β = 28</p><p>α + β = 7</p><p><strong>Step 4:</strong> Substitute β = 4α - 3:</p><p>α + 4α - 3 = 7</p><p>5α = 10</p><p>α = 2, β = 5</p><p><strong>Step 5:</strong> Find radius using point (2, 3):</p><p>r = √[(2 - 2)² + (3 - 5)²] = √[0 + 4] = 2</p><p>∴ Answer: B</p>
Correct Answer: B