Question

Suppose direction cosines of two lines are given by ul+vm+wn=0 and al2+bm2+cn2=0 where u,v,w,a,b,c are arbitrary constants and 1,m,n are direction cosines of the line. The given lines will be perpendicular, if

A
u2(ab)=0
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B
u2(bc)=0
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C
u2(a+b)=0
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D
u2(b+c)=0
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Solution

The correct option is D ∑u2(b+c)=0 Direction cosines of the two lines are given by ul+vm+wn=0 ------(1) and al2+bm2+cn2=0 ------(2) Eliminating n from (1) and (2) gives al2+bm2+c(ul+vm−w)2=0 ⇒w2al2+w2bm2+c(ul+vm)2=0 ⇒(aw2+cu2)(lm)2+2uvc(lm)+(bw2+cv2)=0 l1m1 and l2m2 are roots of above equation. Product of roots l1l2m1m2=bw2+cv2aw2+cu2 -----(3) Similarly, eliminating l from (1) and (2) gives, m1m2n1n2=aw2+cu2av2+bu2 -----(4) From (3) and (4), we get l1l2bw2+cv2=m1m2aw2+cu2=n1n2av2+bu2 Lines are perpendicular if l1l2+m1m2+n1n2=0 ∴u2(b+c)+v2(c+a)+w2(a+b)=0 Hence, option D.

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