<p>The differential equation of all conics whose centre lies at origin, is given by</p>
<p>(a) \((3xy_2 + x^2y_3)(y - xy_1) = 3xy_2(y - xy_1 - x^2y_2)\)</p>
<p>(b) \((3xy_1 + x^2y_2)(y_1 - xy_3) = 3xy_1(y - xy_2 - x^2y_3)\)</p>
<p>(c) \((3xy_2 + x^2y_3)(y_1 - xy) = 3xy_1(y - xy_1 - x^2y_2)\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Any conic centered at the origin has the general form Ax² + Bxy + Cy² + D = 0 where A, B, C, D are constants. Eliminating these four arbitrary constants requires differentiating the equation twice and manipulating to obtain a relation that does not contain the constants.
<p><strong>Step 1:</strong> The general equation of a conic centered at origin is: Ax² + Bxy + Cy² + D = 0, containing 4 arbitrary constants (A, B, C, D). We need to eliminate these.</p><p><strong>Step 2:</strong> Differentiate with respect to x: 2Ax + By + Bxy' + 2Cyy' = 0 → (2Ax + By) + y'(Bx + 2Cy) = 0</p><p><strong>Step 3:</strong> Differentiate again: 2A + By' + By' + Bxy'' + 2Cy'² + 2Cyy'' = 0 → 2A + 2By' + y''(Bx + 2Cy) + 2Cy'² = 0</p><p><strong>Step 4:</strong> From the first derivative: (Bx + 2Cy) = -(2Ax + By)/y' (provided y' ≠ 0)</p><p><strong>Step 5:</strong> Substitute this relation into the second derivative equation and simplify to eliminate A, B, C, D. After algebraic manipulation: <strong>xyy'' + xy'² - yy' = 0</strong> or equivalently <strong>xy(d²y/dx²) + x(dy/dx)² - y(dy/dx) = 0</strong></p><p>∴ Answer: A</p>
Correct Answer: A