<p>The equation of the pair of bisectors of the angles between the pair of lines represented by <br>
\(ax^2 + 2hxy + ay^2 = 0\) <br>
if the product of slopes of four lines represented by a given equation is 1 and a pair of lines represents the bisectors of angles between the other two, then the product of the slopes of each pair is:</p>
Step-by-Step Solution
Key Concept: When two pairs of lines have their angle bisectors coinciding, the slopes satisfy m₁m₂ = m₃m₄ = -1 (perpendicularity condition). The constraint that the product of all four slopes equals 1 forces each pair to have slope product of -1.
<p><strong>Step 1:</strong> Let the four lines have slopes m₁, m₂, m₃, m₄ where the first pair (slopes m₁, m₂) and second pair (slopes m₃, m₄) represent two pairs of lines.</p><p><strong>Step 2:</strong> Given: m₁m₂m₃m₄ = 1</p><p><strong>Step 3:</strong> For angle bisectors of two lines with slopes m₁ and m₂, the bisectors have slopes satisfying the condition that if lines m₃, m₄ are the bisectors, then (m₃ + m₄)/(1 - m₃m₄) relates to m₁ + m₂.</p><p><strong>Step 4:</strong> The angle bisectors of lines with slopes m₁ and m₂ are perpendicular to each other, so if m₃ and m₄ are the bisector slopes: m₃m₄ = -1</p><p><strong>Step 5:</strong> Similarly, for the original pair to have the second pair as their angle bisectors (by symmetry of the configuration): m₁m₂ = -1</p><p><strong>Step 6:</strong> Verify: m₁m₂ · m₃m₄ = (-1)(-1) = 1 ✓ (matches given condition)</p><p><strong>Step 7:</strong> The product of slopes of each pair is m₁m₂ = -1 and m₃m₄ = -1</p><p>∴ Answer: (d) -1</p>
Correct Answer: (d) -1