Let L 1 and L 2 be the following straight lines. L 1 : x − 1 1 = y − 1 = z − 1 3 and L 2 : x − 1 − 3 = y − 1 = z − 1 1 Suppose the straight line L : x − α l = y − 1 m = z − γ − 2 lies in the plane containing L 1 and L 2 , and passes through the point of intersection of L 1 and L 2 . If the line L bisects the acute angle between the lines L 1 and L 2 , then which of the following statements is/are TRUE?
B. l + m = 2
Given the lines L 1 : x − 1 1 = y − 1 = z − 1 3 and L 2 : x − 1 - 3 = y − 1 = z − 1 1 . From the given lines, we can say that both the lines pass through the point 1 ,   0 ,   1 . Now, let a → be the direction vector of line L 1 i.e., a → = i - j + 3 k and let b → be the direction vector of line L 2 i.e., b → = - 3 i - j + k . We also know that the angle bisectors lie in the direction of a → + b →   or   a → - b → . Now, a → . b → = 1 × - 3 + - 1 × - 1 + 3 × 1 = - 3 + 1 + 3 = 1 > 0 . So, direction ratios of the acute angle bisector between two lines will be in the direction of a → + b → i.e., 1 - 3 ,   - 1 - 1 ,   3 + 1 = − 2   ,   − 2   ,   4 = 1 ,   1 ,   - 2 . Since, L : x − α l = y − 1 m = z − γ − 2 is the acute angle bisector between the lines L 1   &   L 2 . So, by comparing the direction ratios, we get l = 1   &   m = 1 ⇒ l + m = 2 So, equation of line L will be x − α 1 = y − 1 1 = z − γ − 2 . Given that the line L passes through the point 1 ,   0 ,   1 . Putting the point in line L , we get 1 − α 1 = 0 − 1 1 = 1 − γ − 2 ⇒ 1 − α = - 1 = 1 − γ − 2 ⇒ 1 − α = - 1   &   1 − γ − 2 = - 1 ⇒ α = 2   &   γ = - 1 ⇒ α − γ = 3 So, options A   &   B are correct.
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