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Q. If the shortest distance between the line joining the points (1,2,3) and (2,3,4), and the line x12=y+11=z20 is α, then 28α2 is equal to

JEE MainJEE Main 2023Three Dimensional Geometry

Solution:

r=(ˆi+2ˆj+3ˆk)+λ(ˆi+ˆj+ˆk)r=a+λp
r=(+ˆiˆj+2ˆk)+μ(2ˆiˆj)r=b+μq
p×q=|ˆiˆjˆk111210|=ˆi+2ˆj3ˆk
d=|(ba)(p×q)|p×q||
d=|(3ˆjˆk)(ˆi+2ˆj3ˆk)14|
=|6+314|=314
α=314
Now, 28α2=282×914=18