The shortest distance between the lines 2x + y + z =1 , 3x + y + 2z = 2 and x = y = z

Q: The shortest distance between the lines 2x + y + z =1 , 3x + y + 2z = 2 and x = y = z

(A) $\frac{1}{\sqrt{2}}$

(B) $\sqrt{2}$

(C) $\frac{3}{\sqrt{2}}$

(D) $\frac{\sqrt{3}}{2}$

Sol. Any plane passing through first line  & if it is parallel to second line

2x + y + z – 1 + λ(3x + y + 2z – 2) = 0,

(2 + 3λ)1 + (1 + λ)1 + (1 + 2λ)1 = 0

⇒ λ = – 2/3.

Plane is y – z + 1 = 0

Distance from (0, 0, 0)  $= \frac{1}{\sqrt{2}}$