Solve: (3x-a)² + (4x-b)² = (x-c)² + (2√6x-d)² with Parameter Constraints

Let's solve the problem by following these detailed steps:

Step 1: Expand Both Sides

• Left Side: $(3x-a)^2 = 9x^2 - 6ax + a^2$
• $(4x-b)^2 = 16x^2 - 8bx + b^2$
• Right Side: $(x-c)^2 = x^2 - 2cx + c^2$
• $(2\sqrt{6}x-d)^2 = 24x^2 - 4\sqrt{6}xd + d^2$

Step 2: Form Complete Expanded Equations

Left Side: $9x^2 - 6ax + a^2 + 16x^2 - 8bx + b^2 = 25x^2 - (6a + 8b)x + (a^2 + b^2)$

Right Side: $x^2 - 2cx + c^2 + 24x^2 - 4\sqrt{6}xd + d^2 = 25x^2 - (2c + 4\sqrt{6}d)x + (c^2 + d^2)$

Step 3: Equate Coefficients

• $x^2$ terms: Coefficient is already checked as equal.
• $- (6a + 8b) = - (2c + 4\sqrt{6}d)$ implies $6a + 8b = 2c + 4\sqrt{6}d$ (1)
• Equate constant terms: $a^2 + b^2 = c^2 + d^2$ (2)

Step 4: Solve the System

• Choose condition $b > 0$ and analyze choice details:
• Using equations and choice alignment, correct viable numbers satisfy both conditions given choice and direct solve constraints.
• Solution: $a = -4\sqrt{6}, b = 3\sqrt{6}, c = -12, d = \sqrt{6}$

Therefore, the values are $a=-4\sqrt{6},b=3\sqrt{6},c=-12,d=\sqrt{6}$.