<p>A circle passes through \((-2, 4)\) and touches the \(y\)-axis at \((0, 2)\). Which one of the following equations can represent a diameter of this circle?</p>
<p>\(2x - 3y + 10 = 0\)</p>
<p>\(3x + 4y - 3 = 0\)</p>
<p>\(4x + 5y - 6 = 0\)</p>
<p>\(5x + 2y + 4 = 0\)</p>
Step-by-Step Solution
Key Concept: If a circle touches the y-axis at point (0,2), its center must lie on the horizontal line y=2 and at distance r from the y-axis. The center is at (r, 2). Use the condition that the circle passes through (-2, 4) to find r, then determine which line passes through both the center and satisfies the diameter condition.
<p><strong>Step 1:</strong> Since the circle touches the y-axis at (0, 2), the center lies on the line y = 2 and the radius equals the horizontal distance from center to y-axis. Let center be C = (a, 2) where a > 0 and radius r = a.</p><p><strong>Step 2:</strong> The circle passes through (-2, 4). Using distance formula: (−2 − a)² + (4 − 2)² = a²</p><p><strong>Step 3:</strong> Expanding: (−2 − a)² + 4 = a² → 4 + 4a + a² + 4 = a² → 4a = −8 → a = −2</p><p><strong>Step 4:</strong> Since a must be positive for the geometry to work (touching at (0,2) from the left), we have center C = (−2, 2) with radius r = 2.</p><p><strong>Step 5:</strong> Any diameter passes through center (−2, 2). The diameter connecting (−2, 4) and (−2, 0) is vertical and passes through the center. The line x = −2 or equations like x + 2 = 0, or lines through (−2, 2) represent diameters.</p><p><strong>Step 6:</strong> The equation passing through the center (−2, 2) in various forms (such as x + 2 = 0, or y − 2 = m(x + 2)) can represent a diameter.</p><p>∴ Answer: A</p>
Correct Answer: A