Ellipse
Standard Form of Ellipse
Grade 11
Question:
<p>An ellipse, with foci at \((0, 2)\) and \((0, -2)\) and minor axis of length 4, passes through which of the following points?</p>
<p>\((\sqrt{2}, 2)\)</p>
<p>\((2, \sqrt{2})\)</p>
<p>\((2, 2\sqrt{2})\)</p>
<p>\((1, 2\sqrt{2})\)</p>
Step-by-Step Solution
Key Concept: Use the definition that sum of distances from any point on ellipse to both foci equals 2a. First find 'a' using the relationship c² = a² - b², where c = 2 (distance from center to focus) and b = 2 (semi-minor axis).
<p><strong>Step 1:</strong> Identify ellipse parameters. Foci at (0, ±2) means c = 2 on y-axis, so major axis is vertical. Minor axis length = 4 means 2b = 4, so b = 2.</p><p><strong>Step 2:</strong> Use c² = a² - b²: 4 = a² - 4, so a² = 8, giving a = 2√2.</p><p><strong>Step 3:</strong> For any point P on the ellipse: |PF₁| + |PF₂| = 2a = 4√2, where F₁ = (0,2) and F₂ = (0,-2).</p><p><strong>Step 4:</strong> Test the given options by calculating the sum of distances to both foci. The point where this sum equals 4√2 lies on the ellipse.</p><p><strong>Step 5:</strong> Verify using the ellipse equation: x²/4 + y²/8 = 1. Substitute each point and check if the equation is satisfied.</p><p>∴ Answer: D</p>
Correct Answer: D