Ellipse
Equation of Ellipse
Grade 11
Question:
<p>The equation of the ellipse whose foci are \((\pm 2, 0)\) and eccentricity is \(1/2\), is</p>
<p>\(\dfrac{x^2}{12} + \dfrac{y^2}{16} = 1\)</p>
<p>\(\dfrac{x^2}{16} + \dfrac{y^2}{12} = 1\)</p>
<p>\(\dfrac{x^2}{16} + \dfrac{y^2}{8} = 1\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: Use the relationship e = c/a and c² = a² - b² to find both semi-major and semi-minor axes from the given foci and eccentricity.
<p><strong>Step 1:</strong> Identify given information. Foci are at (±2, 0), so the major axis is along the x-axis and c = 2.</p><p><strong>Step 2:</strong> Use eccentricity formula: e = c/a. Given e = 1/2 and c = 2:</p><p>1/2 = 2/a ⟹ a = 4</p><p><strong>Step 3:</strong> Find b using the relationship c² = a² - b²:</p><p>4 = 16 - b² ⟹ b² = 12</p><p><strong>Step 4:</strong> Write the standard form of the ellipse with semi-major axis a = 4 along x-axis and b² = 12:</p><p>x²/16 + y²/12 = 1</p><p>∴ Answer: B</p>
Correct Answer: B