<p>The line \(3x + 5y = k\) touches the ellipse \(16x^2 + 25y^2 = 400\) if \(k\) is</p>
Step-by-Step Solution
Key Concept: A line touches an ellipse when the discriminant of the resulting quadratic (after substitution) equals zero. Alternatively, use the condition that for ellipse ax² + by² = 1, line lx + my = n is tangent when l²/a + m²/b = n².
<p><strong>Step 1:</strong> Convert ellipse to standard form: 16x² + 25y² = 400 → x²/25 + y²/16 = 1</p><p><strong>Step 2:</strong> For line 3x + 5y = k to be tangent to ellipse x²/a² + y²/b² = 1, use condition: (l²a² + m²b²) = n², where line is lx + my = n</p><p><strong>Step 3:</strong> Here a² = 25, b² = 16, l = 3, m = 5, n = k</p><p>Apply: 3²(25) + 5²(16) = k²</p><p>9(25) + 25(16) = k²</p><p>225 + 400 = k²</p><p>625 = k²</p><p>k = ±25</p><p><strong>Verification:</strong> Substitute y = (k - 3x)/5 into ellipse equation and verify discriminant = 0 when k = ±25</p><p>∴ Answer: C (k = ±25)</p>
Correct Answer: C