Step 1: Square both sides of the equation to remove the square root. $$C = \sqrt{\frac{U-a}{U-b}}$$ $$C^2 = \left(\sqrt{\frac{U-a}{U-b}}\right)^2$$ $$C^2 = \frac{U-a}{U-b}$$ Step 2: Multiply both sides by $(U-b)$ to eliminate the denominator. $$C^2(U-b) = U-a$$ Step 3: Expand the left side of the equation. $$C^2U - C^2b = U-a$$ Step 4: Collect all terms containing $U$ on one side and terms not containing $U$ on the other side. Subtract $U$ from both sides and add $C^2b$ to both sides. $$C^2U - U = C^2b - a$$ Step 5: Factor out $U$ from the terms on the left side. $$U(C^2 - 1) = C^2b - a$$ Step 6: Divide both sides by $(C^2 - 1)$ to isolate $U$. $$U = \frac{C^2b - a}{C^2 - 1}$$ The final answer is $\boxed{U = \frac{C^2b - a}{C^2 - 1}}$.