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Question

# Suppose the direction cosines of two lines are given by al+bm+cn=0 and fmn+gln+hlm=0 where f,g,h,a,b,c are arbitrary constants and l,m,n are direction cosines of the line. On the basic of the above information and for f=g=h=1 both lines satisfy the relation:

A
a(lm)2+(a+bc)(lm)+b=0
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B
b(mn)2+(b+ca)(mn)+c=0
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C
c(nl)2+(c+ab)nl+a=0
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D
All the above
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Solution

## The correct option is D All the aboveal+bm+cn=0.....(1) ⇒fmn+gln+hlm=0....(2) (1)⇒n=−(al+bmc)⇒(2) ⇒(fm+gl)[−(al+bmc)]+hlm=0 ⇒ag(lm)2+(af+bg−ch)lm+bf=0....(1) if f=g=h=1[a,b,c,d] Similarly, on replacing with m and l, and on simplifyinng, we have the other relations.

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