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Question

If equations of two non intersecting lines are -
xx1l1=yy1m1=zz1n1 & xx2l2=yy2m2=zz2n2 then the shortest distance between them will be -

A
1Σ(l1m2l2m1)2∣ ∣x1y1z1l1m1n1l2m2n2∣ ∣
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B
1Σ(l1m2l2m1)2∣ ∣x2x1y2y1z2z1l1m1n1l2m2n2∣ ∣
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C
1Σ(l1m2l2m1)2∣ ∣x2y2z2l1m1n1l2m2n2∣ ∣
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D
None of these
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Solution

The correct option is B 1Σ(l1m2l2m1)2∣ ∣x2x1y2y1z2z1l1m1n1l2m2n2∣ ∣
Let’s rewrite the first line’s equation -
xx1l1=yy1m1=zz1n1=r1 (say)
Any point on first line( let’s call it “P” ) can be written as (x1+l1.r1,y1+m1.r1,z1+n1.r1)
Similarly equation of second line can be rewritten as -
xx2l2=yy2m2=zz2n2=r2 (say)
And any point on second line( let’s call it “Q” )can be written as
(x2+l2.r2,y2+m2.r2,z2+n2.r2)
Direction ratios of PQ will be -
(x1+l1.r1(x2+l2.r2),y1+m1.r1(y2+m2.r2),z1+n1.r1(z2+n2.r2))
This line will be the shortest line only if it’s perpendicular to both the given lines.
PQ is perpendicular to the first line then -
(x1+l1.r1(x2+l2.r2)).(l1)+(y1+m1.r1(y2+m2.r2))(m1)(z1+n1.r1(z2+n2.r2)).(n1)=0
Similarly, PQ will be perpendicular to the second line -
(x1+l1.r1(x2+l2.r2)).(l2)+(y1+m1.r1(y2+m2.r2))(m2)(z1+n1.r1(z2+n2.r2)).(n2)=0
On solving these equations we get -
1Σ(l1m2l2m1)2∣ ∣x2x1y2y1z2z1l1m1n1l2m2n2∣ ∣

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