If (b+c)x+(c+a)y+(a+b)z=k=(b−c)x+(c−a)y+(a−b)z then what will be the equation of the straight line passing through origin and parallel to given line ?

Krish Crosby 2022-09-23 Answered
If ( b + c ) x + ( c + a ) y + ( a + b ) z = k = ( b c ) x + ( c a ) y + ( a b ) z then what will be the equation of the straight line passing through origin and parallel to given line ?
I tried to relate the direction ratios of the given line and the new line as they are parallel but the problem is that line is passing through origin so I am ending up getting all zeros. Please help. Any hint will do .
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Answers (1)

Kellen Blackburn
Answered 2022-09-24 Author has 8 answers
The given straight line is intersection of the planes P 1 : ( b + c ) x + ( c + a ) y + ( a + b ) z = k and P 2 : ( b c ) x + ( c a ) y + ( a b ) z = k .
Thanks to the coefficients we know that these planes are not parallel. Therefore, the parallel line passing through origin can be defined as intersection of planes parallel to P 1 , P 2 passing through origin. It is
( b + c ) x + ( c + a ) y + ( a + b ) z = 0 = ( b c ) x + ( c a ) y + ( a b ) z .
The vectors normal respectively to P 1 , P 2 represent diagonals of a parallelogram. The sides of this parallelogram are represented by the vectors (b,c,a) and (c,a,b). This is why the planes are not parallel.
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