Ellipse
Normal to Ellipse
Grade 11
Question:
<p>The eccentricity of an ellipse whose centre is at the origin is \(1/2\). If one of its directrices is \(x = -4\), then the equation of the normal to it at \(\left(1, \dfrac{3}{2}\right)\) is</p>
<p>\(4x - 2y = 1\)</p>
<p>\(4x + 2y = 7\)</p>
<p>\(x + 2y = 4\)</p>
<p>\(2y - x = 2\)</p>
Step-by-Step Solution
Key Concept: Use the directrix formula x = -a²/c (where c = ae) along with eccentricity e = 1/2 to find a and b, then apply the normal equation at a point on the ellipse.
<p><strong>Step 1: Find a using directrix and eccentricity</strong></p><p>Directrix: x = -a²/c where c = ae</p><p>So -a²/(ae) = -4 ⟹ a/e = 4</p><p>Given e = 1/2, so a/(1/2) = 4 ⟹ a = 2</p><p><strong>Step 2: Find b</strong></p><p>Since e² = 1 - b²/a² and e = 1/2:</p><p>1/4 = 1 - b²/4 ⟹ b²/4 = 3/4 ⟹ b² = 3</p><p>Ellipse equation: x²/4 + y²/3 = 1</p><p><strong>Step 3: Verify point lies on ellipse</strong></p><p>(1)²/4 + (3/2)²/3 = 1/4 + 9/12 = 1/4 + 3/4 = 1 ✓</p><p><strong>Step 4: Find normal equation at (1, 3/2)</strong></p><p>Normal form: (a²x/x₁) - (b²y/y₁) = a² - b²</p><p>(4·x/1) - (3·y/(3/2)) = 4 - 3</p><p>4x - 2y = 1</p><p><strong>∴ Answer: 4x - 2y - 1 = 0 (or equivalent form)</strong></p>
Correct Answer: D