<p>In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at \((0, 5\sqrt{3})\), then the length of its latus rectum is __________.</p>
Step-by-Step Solution
Key Concept: Since the focus is on the y-axis at (0, 5√3), the major axis is vertical. Use the relationships: 2a - 2b = 10 and c = 5√3, combined with c² = a² - b² for a vertical ellipse to find a and b, then calculate latus rectum = 2b²/a.
<p><strong>Step 1:</strong> Identify the ellipse orientation. Since focus is at (0, 5√3) on the y-axis, the major axis is vertical. The ellipse equation is: x²/b² + y²/a² = 1 where a > b.</p><p><strong>Step 2:</strong> Write the given conditions. We have:</p><ul><li>2a - 2b = 10 ⟹ a - b = 5</li><li>c = 5√3 (distance from center to focus)</li><li>c² = a² - b² (relationship for vertical ellipse)</li></ul><p><strong>Step 3:</strong> From c = 5√3, we get c² = 75. So: a² - b² = 75.</p><p><strong>Step 4:</strong> Solve the system. From a - b = 5, we have a = b + 5. Substitute into a² - b² = 75:</p><p>(b + 5)² - b² = 75</p><p>b² + 10b + 25 - b² = 75</p><p>10b + 25 = 75</p><p>10b = 50</p><p>b = 5</p><p><strong>Step 5:</strong> Therefore a = b + 5 = 10. Verify: a² - b² = 100 - 25 = 75 ✓</p><p><strong>Step 6:</strong> Calculate latus rectum = 2b²/a = 2(25)/10 = 50/10 = 5.</p><p>∴ <strong>Answer: 5</strong></p>
Correct Answer: 5