1. Key Concepts and Formulas

• Direction Cosines: For a line in 3D space, its direction cosines are $l, m, n$, representing the cosines of the angles the line makes with the positive x, y, and z axes, respectively. They satisfy the fundamental property $l^2 + m^2 + n^2 = 1$.
• Direction Ratios: Any set of three numbers $(a, b, c)$ proportional to the direction cosines $(l, m, n)$ are called direction ratios. That is, $l = \frac{a}{\sqrt{a^2+b^2+c^2}}$, $m = \frac{b}{\sqrt{a^2+b^2+c^2}}$, $n = \frac{c}{\sqrt{a^2+b^2+c^2}}$.
• Angle Between Two Lines: If two lines have direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$, the angle $\theta$ between them is given by: $\cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}$ A crucial special case is when the numerator $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$, which implies $\cos \theta = 0$ and thus $\theta = \frac{\pi}{2}$ (the lines are perpendicular).

2. Step-by-Step Solution

Step 1: Identify the Given Equations We are provided with two equations that define the direction cosines $(l, m, n)$ of the lines:

1. $2l + 2m - n = 0 \quad \text{(Equation 1)}$
2. $mn + nl + lm = 0 \quad \text{(Equation 2)}$

• Why this step? These equations represent the constraints on the direction of the lines. Our goal is to solve this system to find the specific direction ratios for each line.

Step 2: Express One Variable in Terms of Others From the simpler linear equation (Equation 1), we can easily express $n$ in terms of $l$ and $m$: $n = 2l + 2m \quad \text{(Equation 3)}$

• Why this step? This substitution strategy reduces the number of variables in the more complex Equation 2, making it easier to solve for the relationships between $l$ and $m$.

Step 3: Substitute and Form a Homogeneous Quadratic Equation Substitute the expression for $n$ from Equation 3 into Equation 2: $m(2l + 2m) + (2l + 2m)l + lm = 0$ Now, expand and simplify the equation: $2lm + 2m^2 + 2l^2 + 2lm + lm = 0$ Combine the like terms: $2l^2 + 5lm + 2m^2 = 0 \quad \text{(Equation 4)}$

• Why this step? This resulting equation is a homogeneous quadratic in $l$ and $m$. Such equations are characteristic in problems involving directions of lines, as they typically yield two distinct ratios of $l$ to $m$, which correspond to the two lines.

Step 4: Solve the Homogeneous Quadratic Equation for the Ratio $l/m$ To solve Equation 4, we can divide the entire equation by $m^2$ (assuming $m \neq 0$).

• Consider $m=0$: If $m=0$, Equation 4 becomes $2l^2 = 0$, implying $l=0$. Substituting $l=0$ and $m=0$ into Equation 3 yields $n = 2(0) + 2(0) = 0$. This would mean $(l,m,n) = (0,0,0)$, which cannot be direction cosines because $l^2+m^2+n^2=1$. Thus, $m$ cannot be zero. Dividing Equation 4 by $m^2$: $2\left(\frac{l}{m}\right)^2 + 5\left(\frac{l}{m}\right) + 2 = 0$ Let $t = \frac{l}{m}$. The equation transforms into a standard quadratic equation in $t$: $2t^2 + 5t + 2 = 0$ Factor the quadratic equation: $2t^2 + 4t + t + 2 = 0$ $2t(t+2) + 1(t+2) = 0$ $(2t+1)(t+2) = 0$ This gives two possible values for $t$: $t = -2 \quad \text{or} \quad t = -\frac{1}{2}$
• Why this step? Each value of $t$ represents a distinct ratio of $l$ to $m$, which defines the orientation of one of the two lines whose direction cosines satisfy the initial conditions.

Step 5: Determine the Direction Ratios for Each Line We use the two ratios obtained for $\frac{l}{m}$ to find the direction ratios $(a, b, c)$ for each line. Since direction ratios are proportional, we can choose a convenient value for one variable (e.g., $m=1$ or $l=1$) to find the others.

Case 1: $\frac{l}{m} = -2$ This implies $l = -2m$. Substitute $l = -2m$ into Equation 3 ($n = 2l + 2m$): $n = 2(-2m) + 2m = -4m + 2m = -2m$ So, for the first line, the direction ratios are proportional to $(l, m, n) = (-2m, m, -2m)$. By setting $m=1$ (or dividing by $m$), we get the direction ratios for the first line: $(a_1, b_1, c_1) = (-2, 1, -2)$

Case 2: $\frac{l}{m} = -\frac{1}{2}$ This implies $m = -2l$. Substitute $m = -2l$ into Equation 3 ($n = 2l + 2m$): $n = 2l + 2(-2l) = 2l - 4l = -2l$ So, for the second line, the direction ratios are proportional to $(l, m, n) = (l, -2l, -2l)$. By setting $l=1$ (or dividing by $l$), we get the direction ratios for the second line: $(a_2, b_2, c_2) = (1, -2, -2)$

• Why this step? We have now successfully derived the direction ratios for the two lines. These sets of numbers $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ precisely define the directions of the lines and are the necessary inputs for calculating the angle between them.

Step 6: Calculate the Angle Between the Two Lines We use the direction ratios for the two lines: Line 1: $(a_1, b_1, c_1) = (-2, 1, -2)$ Line 2: $(a_2, b_2, c_2) = (1, -2, -2)$

The formula for the cosine of the angle $\theta$ between two lines with direction ratios is: $\cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}$

First, calculate the numerator (the dot product of the direction ratio vectors): $a_1 a_2 + b_1 b_2 + c_1 c_2 = (-2)(1) + (1)(-2) + (-2)(-2)$ $= -2 - 2 + 4 = 0$

Next, calculate the denominators (the magnitudes of the direction ratio vectors): $\sqrt{a_1^2+b_1^2+c_1^2} = \sqrt{(-2)^2+1^2+(-2)^2} = \sqrt{4+1+4} = \sqrt{9} = 3$ $\sqrt{a_2^2+b_2^2+c_2^2} = \sqrt{1^2+(-2)^2+(-2)^2} = \sqrt{1+4+4} = \sqrt{9} = 3$

Now, substitute these values into the formula for $\cos \theta$: $\cos \theta = \frac{0}{3 \times 3} = \frac{0}{9} = 0$

Since $\cos \theta = 0$, the angle $\theta$ must be $\frac{\pi}{2}$. $\theta = \cos^{-1}(0) = \frac{\pi}{2}$

• Why this step? This is the final geometric interpretation of our algebraic results. A cosine of 0 indicates that the two lines are mutually perpendicular.

3. Common Mistakes & Tips

• Algebraic Precision: Be extremely careful with signs and calculations during substitution and simplification. A small algebraic error can propagate and lead to an incorrect final answer.
• Homogeneous Quadratic Equations: Recognize that an equation like $2l^2 + 5lm + 2m^2 = 0$ is homogeneous. The standard approach is to divide by $m^2$ (or $l^2$) to form a quadratic in the ratio $l/m$ (or $m/l$). Always consider edge cases like $m=0$ to ensure the division is valid.
• Direction Ratios vs. Direction Cosines: While direction ratios are sufficient for calculating the angle between lines (as the scaling factor cancels out in the formula), remember that actual direction cosines must satisfy $l^2+m^2+n^2=1$. If the question asked for direction cosines, you would need to normalize the ratios.

4. Summary

This problem required us to find the angle between two lines whose direction cosines were defined by a system of two equations. We began by using the linear equation to express one direction cosine in terms of the others, which was then substituted into the quadratic equation. This resulted in a homogeneous quadratic equation in two variables, which we solved for the ratio of those variables. Each ratio yielded a distinct set of direction ratios for the two lines. Finally, we applied the formula for the angle between two lines using their direction ratios, finding that the cosine of the angle was 0, indicating a perpendicular relationship.

5. Final Answer

The final answer is $\boxed{\text{(A)}}$, which corresponds to option (A).