<p>For the ellipse given by \(x^2 + 4y^2 - 2x - 16y + 13 = 0\), which of the following are correct?</p>
<p>(a) Length of latus rectum = 1</p>
<p>(b) Eccentricity \(e = \dfrac{\sqrt{3}}{2}\)</p>
<p>(c) \(2ae = 2\sqrt{3}\)</p>
<p>(d) Length of major axis = 4</p>
Step-by-Step Solution
Key Concept: Convert the general equation to standard form by completing the square, then identify the center, semi-major/minor axes, and use these to verify properties like eccentricity, foci location, and directrix equations.
<p><strong>Step 1: Complete the square</strong></p><p>x² + 4y² - 2x - 16y + 13 = 0</p><p>(x² - 2x) + 4(y² - 4y) + 13 = 0</p><p>(x² - 2x + 1 - 1) + 4(y² - 4y + 4 - 4) + 13 = 0</p><p>(x - 1)² - 1 + 4(y - 2)² - 16 + 13 = 0</p><p>(x - 1)² + 4(y - 2)² = 4</p><p><strong>Step 2: Convert to standard form</strong></p><p>(x - 1)²/4 + (y - 2)²/1 = 1</p><p>Center: (1, 2), a² = 4 (a = 2), b² = 1 (b = 1)</p><p><strong>Step 3: Calculate eccentricity</strong></p><p>c² = a² - b² = 4 - 1 = 3, so c = √3</p><p>e = c/a = √3/2</p><p><strong>Step 4: Find foci</strong></p><p>Foci are at (1 ± √3, 2)</p><p><strong>Step 5: Find directrices</strong></p><p>Directrices: x = 1 ± a/e = 1 ± 2/(√3/2) = 1 ± 4/√3 = 1 ± 4√3/3</p><p>∴ Answer: a, c, d</p>
Correct Answer: a, c, d