    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 l,m,n are direction cosines of the line. The given lines will be parallel if

A
u2(b+c)=0
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B
a2u=0
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C
u2a =0
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D
(b+c)u2=0
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Solution

## The correct option is C ∑u2a =0Direction 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) givesal2+bm2+c(ul+vm−w)2=0⇒w2al2+w2bm2+c(ul+vm)2=0⇒(aw2+cu2)(lm)2+2uvc(lm)+(bw2+cv2)=0l1m1 and l2m2 are roots of above equation, if lines are parallel then direction cosines are equal.i.e discriminant value of above quadratic equation is 0.⇒4u2v2c2=(aw2+cu2)(bw2+cv2)⇒abw4+acw2v2+bcu2w2=0⇒abw2+acv2+bcu2=0∴u2a+v2b+w2c=0Hence, option C.  Suggest Corrections  0      Similar questions  Related Videos   Drawing Tangents to a Circle
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