Ellipse
Equation of Ellipse
Grade 11
Question:
<p>The eccentricity of an ellipse, with its centre at the origin, is \(1/2\). If one of the directrices is \(x = 4\), then the equation of the ellipse is</p>
<p>\(3x^2 + 4y^2 = 1\)</p>
<p>\(3x^2 + 4y^2 = 12\)</p>
<p>\(4x^2 + 3y^2 = 12\)</p>
<p>\(4x^2 + 3y^2 = 1\)</p>
Step-by-Step Solution
Key Concept: Use the directrix formula x = a²/c for an ellipse to find 'a', then apply eccentricity e = c/a to find 'c' and 'b' using b² = a²(1-e²).
<p><strong>Step 1:</strong> For an ellipse with centre at origin and major axis along x-axis, the directrix is given by x = a²/c, where e = c/a.</p><p><strong>Step 2:</strong> Given: e = 1/2 and directrix x = 4, so a²/c = 4.</p><p><strong>Step 3:</strong> From e = c/a = 1/2, we get c = a/2.</p><p><strong>Step 4:</strong> Substitute into a²/c = 4: a²/(a/2) = 4 → 2a = 4 → a = 2. Therefore a² = 4.</p><p><strong>Step 5:</strong> Find c: c = a/2 = 2/2 = 1, so c² = 1.</p><p><strong>Step 6:</strong> Calculate b²: b² = a² - c² = 4 - 1 = 3.</p><p><strong>Step 7:</strong> The equation of the ellipse is x²/4 + y²/3 = 1.</p><p>∴ Answer: B</p>
Correct Answer: B