Ellipse
Equation of Ellipse from eccentricity and directrix
Grade None
Question:
<p>Let the equation of ellipse be \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(e = \frac{1}{2}\) and equation of directrix is \(x = 4\). Then the equation of the ellipse is:</p>
<p>\(3x^2 + 4y^2 = 12\)</p>
<p>\(4x^2 + 3y^2 = 12\)</p>
<p>\(x^2 + 4y^2 = 12\)</p>
<p>\(4x^2 + y^2 = 12\)</p>
Step-by-Step Solution
Key Concept: Use the relationship between eccentricity, semi-major axis, and directrix: directrix x = a/e. From e = 1/2 and x = 4, find a, then use e² = 1 - b²/a² to find b.
<p><strong>Step 1:</strong> For an ellipse with horizontal major axis, the directrix is at x = a/e.</p><p>Given: directrix x = 4 and e = 1/2</p><p>∴ a/e = 4</p><p>a/(1/2) = 4</p><p>2a = 4</p><p>a = 2</p><p><strong>Step 2:</strong> Use the eccentricity formula: e² = 1 - b²/a²</p><p>(1/2)² = 1 - b²/4</p><p>1/4 = 1 - b²/4</p><p>b²/4 = 3/4</p><p>b² = 3</p><p><strong>Step 3:</strong> The equation of the ellipse is:</p><p>x²/4 + y²/3 = 1</p><p>∴ Answer: A</p>
Correct Answer: A