Question

The lines = i – j + l(2i + k) and = (2i – j) + m(i + j – k) intersect for

• A. l = 1, m = 1
• B. l = 2, m = 3
• C. All values of l and m
• D. No value of l and m

Answer 1

Answer: (D)

No value of l and m

Answer 2

The given lines intersect, if the shortest distance between the lines is zero.

We know that the shortest distance between the lines

r = a₁ + (l$\overrightarrow { \mathrm { b } } _ { 1 }$) and r = a₂ + lb₂ is

$\frac { \left| \left( a _ { 1 } - a _ { 2 } \right) \cdot b _ { 1 } \times b _ { 2 } \right| } { \left| b _ { 1 } \times b _ { 2 } \right| }$

So the shortest distance between the given lines is zero if

(i – j – (2i – j) . (2i + k) × (i + j – k) = 0

L.H.S. =

$\left| \begin{array} { c c c } - 1 & 0 & 0 \\ 2 & 0 & 1 \\ 1 & 1 & - 1 \end{array} \right|$ = 1 ¹ 0

Hence the given lines do not intersect.