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If the lines (x-2)/(1)=(y-3)/(1)=(z-4)...

If the lines ` (x-2)/(1)=(y-3)/(1)=(z-4)/(-k) ` and ` (x-1)/(k)=(y-4)/(2)=(z-5)/(1) ` are coplanar, then find the value of k.

A

any value

B

exactly one value

C

exactly two values

D

exactly three values

Text Solution

Verified by Experts

The correct Answer is:
C
Condition for two lines are coplanar.
`|{:(x_(1)-x_(2),y_(1)-y_(2),z_(1)-z_(2)),(" "l_(1)," "m_(1)," "n_(1)),(" "l_(2)," "m_(2)," "n_(2)):}|=0`
where, `(x_(1), y_(a), z_(1))` and `(x_(2), y_(2), z_(2))` are the points lie on lines (i) and (ii) respectively and `ltl_(1), m_(1), n_(1)gt` and `ltl_(2), m_(2), n_(2)gt` are the dirction cosines of the lines (i) and (ii), respectively.
`:." "|{:(2-1,3-4,4-5),(" "1," "1," "-k),(" "k," "2," "1):}|=0`
`implies" "|{:(1,-1," "-1),(1," "1," "-k),(k," "2," "1):}|=0`
`implies1(1+2k)+(1+k^(2))-(2-k)=0`
`implies" "k^(2)+2k+k=0`
`implies" "k^(2)+3k=0`
`implies" "k=0,-3`
If 0 appears in the denominator, then the correct way of representing the equation of straight line is
`(x-2)/(1)=(y-3)/(1),z=4`
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