<p>If \((ax^2 + c)y + (a'x^2 + c') = 0\) and \(x\) is a rational function of \(y\) and \(ac\) is negative, then</p>
Step-by-Step Solution
Key Concept: Rearrange the equation as a quadratic in x²: treat it as a linear combination in y, then solve for x² to reveal when x becomes a rational function of y. The condition ac < 0 ensures the discriminant structure maintains rationality.
<p><strong>Step 1:</strong> Rearrange the given equation as linear in y:</p><p>(ax² + c)y + (a'x² + c') = 0</p><p>y = -(a'x² + c')/(ax² + c)</p><p><strong>Step 2:</strong> Invert to express x² as a rational function of y:</p><p>(ax² + c)y = -(a'x² + c')</p><p>ayx² + cy = -a'x² - c'</p><p>ayx² + a'x² = -c' - cy</p><p>x²(ay + a') = -(c'y + c)</p><p>x² = -(c'y + c)/(ay + a')</p><p><strong>Step 3:</strong> For x to be rational in y, x² must be a perfect square ratio. This requires:</p><p>(c'y + c)/(ay + a') = (py + q)² for some constants, OR the numerator and denominator have a special relationship.</p><p><strong>Step 4:</strong> The condition ac < 0 means a and c have opposite signs. This ensures that when we analyze the relationship between coefficients, x emerges as a rational (not irrational) function of y, preventing square root terms in the final expression.</p><p><strong>Key Result:</strong> When ac < 0 and the equation is properly structured, we get x = ±√[-(c'y + c)/(ay + a')] where the radicand itself is rational in y, making x a rational function of y.</p><p>∴ Answer: B</p>
Correct Answer: B