Question

If the direction ratios of two lines are given by \[ l+m+n=0 \] \[ mn-2ln+lm=0 \] then the angle between the lines is

• A. \(\dfrac{\pi}{4}\)
• B. \(\dfrac{\pi}{3}\)
• C. \(\dfrac{\pi}{2}\)
• D. \(0\)

Hint:

If \[ \cos\theta=0 \] then the two lines are perpendicular.

Answer 1

The Correct Option is C

Concept:
Use the equations to find the direction ratios and then determine the angle between lines.

Step 1: Use the first equation. \[ l+m+n=0 \] \[ n=-(l+m) \] Substitute into second equation.

Step 2: Substitute into the second equation. Given, \[ mn-2ln+lm=0 \] Putting \(n=-(l+m)\), \[ m[-(l+m)]-2l[-(l+m)]+lm=0 \] \[ -lm-m^2+2l^2+2lm+lm=0 \] \[ 2l^2+2lm-m^2+lm=0 \] \[ 2l^2+2lm-m^2=0 \]

Step 3: Find possible ratios. Let, \[ l=1 \] Then, \[ 2+2m-m^2=0 \] \[ m^2-2m-2=0 \] Solving, \[ m=1\pm\sqrt3 \] Corresponding direction ratios give two lines.

Step 4: Find the angle between lines. Using formula, \[ \cos\theta= \frac{l_1l_2+m_1m_2+n_1n_2} {\sqrt{l_1^2+m_1^2+n_1^2}\sqrt{l_2^2+m_2^2+n_2^2}} \] After simplification, \[ \cos\theta=0 \] Therefore, \[ \theta=\frac{\pi}{2} \]

Step 5: Conclusion. \[ \boxed{\theta=\frac{\pi}{2}} \] Hence the correct answer is: \[ \boxed{(C)} \]