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QJEEM22Ma8d57: JEE Main 2022 Mathematics - Class 12 JEEM22Ma8d57 Solution: If the shortest distance between the lines r → = ( − i ^ + 3 k…

If the shortest distance between the lines

r→=(−i^+3k^)+λ(i^−aj^)

and r→=(−j^+2k^)+μ(i^−j^+k^) is 23, then the integral value of a is equal to ___________.

Step-by-Step Verified Solution

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a⃗1−a⃗2=−i^+j^+k^\vec{a}_{1}-\vec{a}_{2}=-\hat{i}+\hat{j}+\hat{k}

Shortest distance =∣(a⃗1−a⃗2)⋅(b⃗1×b⃗2)∣b⃗1×b⃗2∣∣=\left|\frac{\left(\vec{a}_{1}-\vec{a}_{2}\right) \cdot\left(\vec{b}_{1} \times \vec{b}_{2}\right)}{\left|\vec{b}_{1} \times \vec{b}_{2}\right|}\right|

⇒23=2(a−1)a2+1+(a−1)2⇒6(a2−2a+1)=2a2−2a+2⇒(a−2)(2a−1)=0⇒a=2 because a∈z.

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