1. Key Concepts and Formulas

• Direction Cosines: For a line in 3D space, its direction cosines ($l, m, n$) are the cosines of the angles it makes with the positive x, y, and z axes, respectively. They satisfy the fundamental relation $l^2 + m^2 + n^2 = 1$.
• Parallel Lines: Two lines are parallel if and only if they have the same direction ratios (or direction cosines). When a system of equations defines the direction cosines of "two lines" that are stated to be parallel, it implies that the system must yield a unique set of direction ratios $(l:m:n)$.
• Quadratic Equation and Equal Roots: A quadratic equation of the form $Ax^2 + Bx + C = 0$ has equal roots if and only if its discriminant, $D = B^2 - 4AC$, is equal to zero.

2. Step-by-Step Solution

Step 1: State the Given Relations We are given two algebraic relations involving the direction cosines ($l, m, n$) of the lines: $l + m - n = 0 \quad \text{(Equation 1)}$ $3l^2 + m^2 + cnl = 0 \quad \text{(Equation 2)}$

Step 2: Eliminate One Variable Using the Linear Relation Our goal is to find the direction ratios $(l:m:n)$ that satisfy both equations. Equation 1 is a linear relation, which is ideal for substitution to reduce the number of variables. From Equation 1, we can express $n$ in terms of $l$ and $m$: $n = l + m \quad \text{(Equation 3)}$ Reasoning: By substituting $n$ from Equation 3 into Equation 2, we transform the problem into an equation involving only two variables, $l$ and $m$, which simplifies finding their ratio.

Step 3: Substitute and Simplify the Quadratic Relation Now, substitute Equation 3 into Equation 2: $3l^2 + m^2 + c(l + m)l = 0$ Expand and simplify the expression: $3l^2 + m^2 + cl^2 + clm = 0$ Group the terms by $l^2$, $lm$, and $m^2$: $(3 + c)l^2 + clm + m^2 = 0 \quad \text{(Equation 4)}$ Reasoning: This is a homogeneous quadratic equation in $l$ and $m$. Such equations are crucial because they directly allow us to determine the ratio of the variables, which represents the direction ratios of the line.

Step 4: Apply the Condition for Parallel Lines (Equal Roots) The problem states that the two lines are parallel. This is the key piece of information. If there were two distinct lines, Equation 4 would typically yield two different ratios of $l/m$. To see this, we can divide Equation 4 by $m^2$ (assuming $m \neq 0$). This transforms the equation into a quadratic in the ratio $l/m$: $(3 + c)\left(\frac{l}{m}\right)^2 + c\left(\frac{l}{m}\right) + 1 = 0$ Let $k = \frac{l}{m}$. Then the equation becomes a standard quadratic equation in $k$: $(3 + c)k^2 + ck + 1 = 0 \quad \text{(Equation 5)}$ Reasoning: Since the lines are parallel, they share the same direction. This means there must be only one unique ratio $l/m$ (or $k$) that satisfies the conditions. For a quadratic equation to have only one unique solution (i.e., equal roots), its discriminant must be zero. (Note: The case $m=0$ means $k$ is infinite. For $k$ to be uniquely infinite, the coefficient of $k^2$ must be zero, but then the remaining linear equation must have a unique solution. However, the discriminant condition is the general method for ensuring a unique finite root, which is sufficient here).

Step 5: Use the Discriminant Condition for Equal Roots For a quadratic equation $Ax^2 + Bx + C = 0$ to have equal roots, its discriminant $D = B^2 - 4AC$ must be zero. In Equation 5, we have $A = (3 + c)$, $B = c$, and $C = 1$. Setting the discriminant to zero: $c^2 - 4(3 + c)(1) = 0$ $c^2 - 4(3 + c) = 0$ Reasoning: This algebraic condition directly ensures that Equation 5 yields only one possible value for $k$, which corresponds to a single, unique direction for the parallel lines.

Step 6: Solve for c Simplify and solve the quadratic equation for $c$: $c^2 - 12 - 4c = 0$ $c^2 - 4c - 12 = 0$ Factor the quadratic equation: $(c - 6)(c + 2) = 0$ This gives two possible values for $c$: $c = 6 \quad \text{or} \quad c = -2$ The question asks for the positive value of $c$. Therefore, $c = 6$.

3. Common Mistakes & Tips

• Understanding "Parallel Lines": The most crucial step is understanding that "two lines are parallel" in this context implies that the system of equations for their direction cosines must yield a unique set of direction ratios. This directly translates to the discriminant of the resulting quadratic equation being zero.
• Homogeneous Equations: Recognize that homogeneous quadratic equations in two variables (like Equation 4) can always be converted into a quadratic equation in the ratio of the variables by dividing by one of the variables squared.
• Checking for $m=0$: While we divided by $m^2$ assuming $m \neq 0$, the discriminant method is robust. If $m=0$ were the unique valid case, it would mean $l \neq 0$ (as $l,m,n$ cannot all be zero). This implies $k = l/m$ would be infinite. For a quadratic $Ak^2+Bk+C=0$ to have a single infinite root, we'd need $A=0$ and $B \neq 0$. So $3+c=0 \Rightarrow c=-3$, and $c \neq 0$. If $c=-3$, Equation 5 becomes $-3k+1=0$, which gives $k=1/3$, meaning $l/m=1/3$. This contradicts $m=0$. Thus, the assumption $m \neq 0$ and the discriminant approach is valid for finding a unique finite ratio.
• Positive Value: Always double-check the question for specific constraints, such as "positive value" or "negative value," to select the correct answer from multiple possibilities.

4. Summary

The problem involves finding a parameter 'c' such that two lines, whose direction cosines are defined by two given relations, are parallel. The condition for parallel lines means that there should be only one unique direction satisfying the given relations. By substituting the linear relation into the quadratic one, we obtained a homogeneous quadratic equation in $l$ and $m$. Dividing this by $m^2$ yielded a quadratic equation in the ratio $l/m$. For a unique direction, this quadratic must have equal roots, which implies its discriminant must be zero. Solving the resulting equation for $c$ gave $c=6$ or $c=-2$. Since the question asks for the positive value of $c$, the answer is 6.

5. Final Answer The final answer is $\boxed{6}$, which corresponds to option (A).