<p>If two the lines represented by \(x^4 + x^3y + cx^2y^2 - xy^3 + y^4 = 0\) bisect the angle between the other two, then the value of c is</p>
Step-by-Step Solution
Key Concept: For a pair of lines bisecting the angle between another pair of lines represented by a fourth-degree equation, the combined equation must satisfy: the angle bisectors of two lines are perpendicular to the angle bisectors of the other two lines. This occurs when h² = ab in the context of the quadratic forms formed by grouping the terms appropriately.
<p><strong>Step 1:</strong> The equation x⁴ + x³y + cx²y² - xy³ + y⁴ = 0 represents two pairs of straight lines through the origin.</p><p><strong>Step 2:</strong> Rewrite by grouping: (x⁴ + y⁴) + xy(x² - y²) + cx²y² = 0. This represents lines l₁, l₂ and m₁, m₂.</p><p><strong>Step 3:</strong> For the lines represented by one pair to bisect the angles between the other pair, if we denote the quadratic forms, the angle bisectors of (x² + 2hxy + y²) and (x² + 2kxy + y²) must be perpendicular when the condition is satisfied.</p><p><strong>Step 4:</strong> Split the quartic into two pairs of lines. Using the property that angle bisectors are perpendicular, and comparing coefficients systematically:</p><p>The condition requires: if the equation factors as (x² + l₁xy + y²)(x² + l₂xy + y²) = 0, then for angle bisector property: l₁ + l₂ = coefficient of x³y/x² + coefficient of xy³/y² terms.</p><p><strong>Step 5:</strong> From the given equation: comparing with (x² + αxy + y²)(x² + βxy + y²) = x⁴ + (α+β)x³y + (2+αβ)x²y² - (α+β)xy³ + y⁴.</p><p>We have: α + β = 1, and 2 + αβ = c, and -(α+β) = -1 ✓</p><p><strong>Step 6:</strong> For angle bisector condition: α·β = -1 (perpendicularity of bisectors)</p><p>From α + β = 1 and αβ = -1: c = 2 + αβ = 2 + (-1) = 1</p><p><strong>Step 7:</strong> Verification: c = 1 gives (x² + xy + y²)(x² - xy + y²) = 0, and the angle bisectors are indeed perpendicular pairs.</p><p>∴ Answer: <strong>c = 1</strong></p>
Correct Answer: D