<p>The equation of bisectors of lines <em>xy</em> = 0 are <em>y</em> = ±<em>x</em>. If these lines are contained in the pair of lines
\[my^2 + (1 - m^2)xy - mx^2 = 0,\]
then what is the value of <em>m</em>?</p>
Step-by-Step Solution
Key Concept: If y = x and y = -x are both contained in the pair of lines, then they must satisfy the equation identically. Since these are the angle bisectors of xy = 0, the given pair must represent y = x and y = -x as its two component lines.
<p><strong>Step 1:</strong> If y = x and y = -x are the two lines in the pair, then the equation must be expressible as:</p><p>k(y - x)(y + x) = 0 for some constant k</p><p>k(y² - x²) = 0</p><p>ky² - kx² = 0</p><p><strong>Step 2:</strong> Expand (y - x)(y + x) = y² - x². Compare with my² + (1 - m²)xy - mx² = 0:</p><p>The coefficient of xy in the factored form is 0, so we need: 1 - m² = 0</p><p>This gives m² = 1, so m = ±1</p><p><strong>Step 3:</strong> Check which value works. The equation becomes:</p><p>For m = 1: y² + (1 - 1)xy - x² = y² - x² ✓ (represents y = x and y = -x)</p><p>For m = -1: -y² + (1 - 1)xy + x² = -y² + x² = -(y² - x²) ✓ (also represents y = ±x)</p><p>Both m = 1 and m = -1 work, but typically m = 1 is taken as the answer.</p><p>∴ Answer: <strong>m = 1</strong> (or m = -1)</p>
Correct Answer: B