Question

If the direction cosines of two lines satisfy the relations \[ l+m-n=0, \] \[ 3l^2+m^2+6nl=0, \] then those lines are

• A. parallel lines
• B. perpendicular lines
• C. skew lines
• D. intersecting lines

Hint:

If two lines have proportional direction ratios (or direction cosines differing only by sign), then they are \[ \boxed{\text{parallel}.} \] Always reduce the given relations to obtain the ratio of the direction cosines.

Answer 1

The Correct Option is A

Step 1: Express one direction cosine in terms of the others. From \[ l+m-n=0, \] we get \[ n=l+m. \] Substituting into \[ 3l^2+m^2+6nl=0, \] gives \[ 3l^2+m^2+6l(l+m)=0, \] or \[ 9l^2+6lm+m^2=0. \]

Step 2: Solve for the ratio of direction cosines. Observe that \[ 9l^2+6lm+m^2=(3l+m)^2. \] Hence, \[ 3l+m=0, \] so \[ m=-3l. \] Also, \[ n=l+m=l-3l=-2l. \] Thus, \[ l:m:n=1:-3:-2. \]

Step 3: Draw the conclusion. Since the direction cosines are uniquely determined (up to sign), both lines have the same direction ratios. Hence, the two lines are \[ \boxed{\text{parallel lines}.} \] Thus, \[ \boxed{(A)} \] is the correct answer.