Verified Solution • 2025–26 Session moderate

QJEEM21Ma4559: JEE Main 2021 Mathematics - Class 12 JEEM21Ma4559 Solution: Let be an integer. If the shortest distance between the lines x …

Let λ\lambda be an integer. If the shortest distance between the lines

x −- λ\lambda = 2y −- 1 = −- 2z and x = y + 2 λ\lambda = z −- λ\lambda is 722{{\sqrt 7 } \over {2\sqrt 2 }} , then the value of | λ\lambda | is _________.

Step-by-Step Verified Solution

x−λ1=y−1212=z−12{{x - \lambda } \over 1} = {{y - {1 \over 2}} \over {{1 \over 2}}} = {z \over { - {1 \over 2}}}

x−λ2=y−121=2−1{{x - \lambda } \over 2} = {{y - {1 \over 2}} \over 1} = {2 \over { - 1}} ....... (1)

Point on line = (λ,12,0)\left( {\lambda ,{1 \over 2},0} \right)

x1=y+2λ1=z−λ1{x \over 1} = {{y + 2\lambda } \over 1} = {{z - \lambda } \over 1} ....... (2)

Point on line = (0,−2λ,λ)(0, - 2\lambda ,\lambda )

Distance between skew lines =[a→2−a→1b→1b→2]∣b→1×b→2∣= {{\left[ {{{\overrightarrow a }_2} - {{\overrightarrow a }_1}{{\overrightarrow b }_1}{{\overrightarrow b }_2}} \right]} \over {\left| {{{\overrightarrow b }_1} \times {{\overrightarrow b }_2}} \right|}}

|λ12+2λ−λ21−1111|

|i^j^k^21−1111|―

=∣−5λ−32∣14=722= {{\left| { - 5\lambda - {3 \over 2}} \right|} \over {\sqrt {14} }} = {{\sqrt 7 } \over {2\sqrt 2 }}

=∣10λ+3∣=7⇒λ=−1= |10\lambda + 3| = 7 \Rightarrow \lambda = - 1

⇒∣λ∣=1\Rightarrow |\lambda | = 1

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