Ellipse
Tangent to Ellipse
Grade 11
Question:
<p>If the ordinate of the point of contact be 2 then the equation of the tangent to \(x^2 + 4y^2 = 25\) is</p>
<p>(a) \(3x + 8y = 25\)</p>
<p>(b) \(8x + 3y = 25\)</p>
<p>(c) \(8y - 3x = 25\)</p>
<p>(d) \(3x - 8y = 25\)</p>
Step-by-Step Solution
Key Concept: Use the ordinate (y-coordinate) value to find the corresponding x-coordinate on the ellipse, then apply the tangent equation formula for ellipses: xx₁/a² + yy₁/b² = 1.
<p><strong>Step 1:</strong> Given ellipse: x² + 4y² = 25. Rewrite as x²/25 + y²/(25/4) = 1, so a² = 25, b² = 25/4.</p><p><strong>Step 2:</strong> Point of contact has ordinate y₁ = 2. Substitute into ellipse equation: x² + 4(2)² = 25 → x² + 16 = 25 → x² = 9 → x₁ = ±3.</p><p><strong>Step 3:</strong> Use tangent formula at point (x₁, y₁): xx₁/a² + yy₁/b² = 1.</p><p><strong>Step 4:</strong> For point (3, 2): (x·3)/25 + (y·2)/(25/4) = 1 → 3x/25 + 8y/25 = 1 → 3x + 8y = 25.</p><p><strong>Step 5:</strong> For point (-3, 2): (x·(-3))/25 + (y·2)/(25/4) = 1 → -3x/25 + 8y/25 = 1 → -3x + 8y = 25 or 3x - 8y = -25.</p><p>∴ Answer: A (The tangent equations are 3x + 8y = 25 or 3x - 8y = -25)</p>
Correct Answer: A