Question

The equation of the plane containing the lines \(\frac{x-1}{2} = \frac{y+1}{λ} = \frac{z}{2}\) and \(\frac{x+1}{5} = \frac{y+1}{2} = \frac{z}{λ}\) is

• A. \(x\pm y+1 = 0\)
• B. \(y\pm z+1 = 0\)
• C. \(x\pm z+1 = 0\)
• D. \(y\pm z-1 = 0\)

Hint:

Coplanarity gives lambda = plus or minus 2. Then cross the directions.

Answer 1

The Correct Option is B

Step 1: Common feature:
Both lines have the same y-value at their given points, $y=-1$, and the same $z=0$.

Step 2: Check option (B) for lambda = 2:
Direction $(2,2,2)$ with the plane $y-z+1=0$: normal $(0,1,-1)$ dot $(2,2,2)=0$. Point $(1,-1,0)$: $-1-0+1=0$. Good. Line 2 direction $(5,2,2)$: dot $=0$. Point $(-1,-1,0)$: $0$. Good.

Step 3: Check others:
Option (A) $x\pm y+1=0$ at $(1,-1,0)$ gives $1-1+1=1\ne0$ or $1+1+1\ne0$. Option (C) $x\pm z+1=0$ at $(1,-1,0)$ gives $2\ne0$. Option (D) $y\pm z-1=0$ at $(1,-1,0)$ gives $-2\ne0$. So only (B).

Final Answer:
Option B passes through both given points for lambda = plus or minus 2. \[ \boxed{B} \]