the lines `(x-2)/1` = `(y-3)/1` = `(z-4)/-k` and `(x-1)/k` = `(y-4)/1` = `(z-5)/1` are coplanar if k=?

Options

A. any value
B. exactly one value
C. exactly two values
D. exactly three values

Answer

C

Explanation

Condition for two lines to be coplanar.
` [ a- c, b, d] = 0 `
where, a and c are the points on a line and b and d are the direction ratios of the lines
` therefore |{:(2 - 1 , 3 - 4, 4 - 5 ) , ( 1 , 1 , - k ) , ( k , 2 , 1 ) :}| = 0 rArr |{:( 1, - 1, - 1 ) , ( 1 , 1 , k ) , ( k , 2 , 1 ) :} | = 0 `
` rArr 1 ( 1 + 2 k) + ( 1 + k ^ 2 ) - ( 2 - k ) = 0 `
` rArr k ^ 2 + 2k + k = 0 `
` rArr k ^ 2 + 3k = 0 rArr k = 0 , - 3 `
Therefore k have two values.