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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 basis 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 of the above.
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Solution

The correct option is D All of the above.
al+bm+cn=0.....(1)
fmn+gln+hlm=0....(2)
From (1)n=(al+bmc)
Putting in (2), we get
(fm+gl)[(al+bmc)]+hlm=0
ag(lm)2+(af+bgch)lm+bf=0....(1)
if f=g=h=1
a(lm)2+(a+bc)(lm)+b=0

From (1)m=(al+cnb)
Putting in (2), we get
If f=g=h=1
c(nl)2+(c+ab)nl+a=0

From (1)l=(cn+bma)
Putting in (2), we get
if f=g=h=1
b(mn)2+(b+ca)(mn)+c=0

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