<p>If the pair of straight lines \(x^2 - 2pxy - y^2 = 0\) and \(x^2 - 2qxy - y^2 = 0\) be such that each pair bisects the angle between the other pair, then</p>
Step-by-Step Solution
Key Concept: If two pairs of lines bisect each other's angles, their combined equations must represent four lines through origin where adjacent pairs are perpendicular. This occurs when the two pairs are symmetric about perpendicular directions, leading to the constraint pq = -1.
<p><strong>Step 1:</strong> The pair of lines <strong>x² - 2pxy - y² = 0</strong> has slopes m₁, m₂ satisfying: m₁ + m₂ = 2p and m₁m₂ = -1.</p><p><strong>Step 2:</strong> The pair of lines <strong>x² - 2qxy - y² = 0</strong> has slopes m₃, m₄ satisfying: m₃ + m₄ = 2q and m₃m₄ = -1.</p><p><strong>Step 3:</strong> For the first pair to bisect angles of the second pair, the slopes must be related such that bisectors of one pair are perpendicular to the other. If m₁, m₃ are adjacent slopes, the angle bisector condition requires:</p><p><strong>Step 4:</strong> The slopes of angle bisectors of pair 1 are: tan(θ₁ ± π/4) which relates to ±1/(2p) slope direction. Similarly for pair 2: ±1/(2q).</p><p><strong>Step 5:</strong> For mutual angle bisection: the bisectors must coincide, requiring the pairs to be perpendicular positioned. This gives: <strong>pq = -1</strong></p><p><strong>Step 6:</strong> Verification: If pq = -1, then (2p)(2q) = 4pq = -4, ensuring the four lines form a configuration where each pair symmetrically bisects the other.</p><p>∴ Answer: <strong>pq = -1</strong> (Option D)</p>
Correct Answer: D