If the ratio of the roots of the equation x2- 2ax + b = 0 is equal to that of the roots x2 -2cx + d = 0, then find which option is correct. यदि समीकरण x2- 2ax + b = 0 के मूल और समीकरण x2 -2cx + d = 0 के मूल बराबर हो, तो कौन सा विकल्प सही होगा|

For quadratic ax² + bx + c = 0 with roots α, β: sum = α+β = -b/a and product = αβ = c/a. Equation 1: x² - 2ax + b = 0 has sum = 2a, product = b. Equation 2: x² - 2cx + d = 0 has sum = 2c, product = d. Given ratios are equal: α/β = p/q (say). Then α²/αβ = p²/(pq). But αβ = product, so α² = (sum/product) × product²? Let's use α/β = k. Then α = kβ and αβ = kβ² = product. Also α+β = kβ+β = β(k+1) = sum. For ratio equality: α₁/β₁ = α₂/β₂ = k. Then α₁β₁ = kβ₁² = b, α₂β₂ = kβ₂² = d. Also: (α₁+β₁)² = β₁²(k+1)² = (2a)² = 4a². So β₁²(k+1)² = 4a² and kβ₁² = b. Dividing: (k+1)²/k = 4a²/b. Similarly: (k+1)²/k = 4c²/d. Equating: 4a²/b = 4c²/d, so a²/b = c²/d, giving a²d = c²b.