<p>If the roots of \((a^2 + b^2)x^2 - 2(bc + ad)x + c^2 + d^2 = 0\) are equal, then</p>
<p>(a) \(\frac{a}{b} = \frac{c}{d}\)</p>
<p>(b) \(\frac{a}{c} + \frac{b}{d} = 0\)</p>
<p>(c) \(\frac{a}{d} = \frac{c}{d}\)</p>
<p>(d) a + b = c + d</p>
Step-by-Step Solution
Key Concept: A quadratic equation has equal roots if and only if its discriminant equals zero. Apply this condition to the given equation and simplify the resulting expression using algebraic identities.
<p><strong>Step 1:</strong> For a quadratic equation Ax² + Bx + C = 0 to have equal roots, the discriminant Δ = B² - 4AC = 0.</p><p><strong>Step 2:</strong> In our equation, A = a² + b², B = -2(bc + ad), and C = c² + d². Apply the discriminant condition:</p><p>[-2(bc + ad)]² - 4(a² + b²)(c² + d²) = 0</p><p><strong>Step 3:</strong> Expand the first term:</p><p>4(bc + ad)² - 4(a² + b²)(c² + d²) = 0</p><p><strong>Step 4:</strong> Divide by 4:</p><p>(bc + ad)² - (a² + b²)(c² + d²) = 0</p><p><strong>Step 5:</strong> Expand (bc + ad)²:</p><p>b²c² + 2abcd + a²d² - (a²c² + a²d² + b²c² + b²d²) = 0</p><p><strong>Step 6:</strong> Simplify:</p><p>b²c² + 2abcd + a²d² - a²c² - a²d² - b²c² - b²d² = 0</p><p>2abcd - a²c² - b²d² = 0</p><p><strong>Step 7:</strong> Rearrange:</p><p>a²c² - 2abcd + b²d² = 0</p><p>(ac - bd)² = 0</p><p><strong>Step 8:</strong> This gives us ac - bd = 0, or ac = bd, which means:</p><p>a/b = c/d</p><p><strong>∴ Answer:</strong> a</p>
Correct Answer: a