Question

The points (0, $\lambda$, 1), ($\mu$, 3, -1), ($\lambda$, 5, 0), ($\mu$, 6, $\mu$) taken in that order, form a square. If $\lambda$, $\mu$ are positive real numbers, then the length of its side is

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For problems with squares in 3D space, equating the midpoints of the diagonals is usually the fastest way to solve for unknown variables.

Answer 1

The Correct Option is C

Step 1: Concept
If four points $A, B, C, D$ form a square, then the lengths of all sides must be equal ($AB = BC = CD = DA$) and the diagonals must be equal ($AC = BD$). Also, the midpoints of both diagonals must coincide.

Step 2: Meaning
Let $A=(0, \lambda, 1)$, $B=(\mu, 3, -1)$, $C=(\lambda, 5, 0)$, and $D=(\mu, 6, \mu)$. The midpoint of diagonal $AC$ is $\left(\frac{\lambda}{2}, \frac{\lambda+5}{2}, \frac{1}{2}\right)$, and the midpoint of $BD$ is $\left(\frac{2\mu}{2}, \frac{3+6}{2}, \frac{-1+\mu}{2}\right) = \left(\mu, 4.5, \frac{\mu-1}{2}\right)$.

Step 3: Analysis
Equating the midpoints component-wise: 1. From the y-coordinate: $\frac{\lambda+5}{2} = 4.5 \implies \lambda + 5 = 9 \implies \lambda = 4$. 2. From the x-coordinate: $\mu = \frac{\lambda}{2} = \frac{4}{2} = 2$. Let's verify with the z-coordinate: $\frac{\mu-1}{2} = \frac{2-1}{2} = \frac{1}{2}$, which perfectly matches. Thus, $\lambda = 4$ and $\mu = 2$. The points are $A(0, 4, 1)$ and $B(2, 3, -1)$.

Step 4: Conclusion
The length of side $AB$ is computed using the distance formula: $AB = \sqrt{(2-0)^2 + (3-4)^2 + (-1-1)^2} = \sqrt{2^2 + (-1)^2 + (-2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3$.

Final Answer: (C)