Key Concepts and Formulas

1. Equation of a Line (Symmetric Form): The equation of a line passing through a point $(x_1, y_1, z_1)$ and having direction ratios $(a, b, c)$ is given by: $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$
2. General Point on a Line (Parametric Form): Any point on a line represented as $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$ can be expressed parametrically by equating it to a scalar parameter, say $t$: $x = x_1 + at, \quad y = y_1 + bt, \quad z = z_1 + ct$
3. Condition for a Line Parallel to a Plane: If a line with direction vector $\vec{d} = a\hat{i} + b\hat{j} + c\hat{k}$ is parallel to a plane with normal vector $\vec{n} = A\hat{i} + B\hat{j} + C\hat{k}$ (where $A, B, C$ are coefficients of $x, y, z$ in the plane equation $Ax+By+Cz+D=0$), then the direction vector of the line must be perpendicular to the normal vector of the plane. This implies their dot product is zero: $\vec{d} \cdot \vec{n} = 0 \quad \Rightarrow \quad aA + bB + cC = 0$

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Step-by-Step Solution

Step 1: Define the Equation of the Required Line $L$

We are given that the required line $L$ passes through the point $A(-4, 3, 1)$. Let the direction ratios of this line be $(a, b, c)$. Using the symmetric form of a line, its equation can be written as: $\frac{x - (-4)}{a} = \frac{y - 3}{b} = \frac{z - 1}{c} \quad \Rightarrow \quad \frac{x + 4}{a} = \frac{y - 3}{b} = \frac{z - 1}{c}$ Our primary goal is to determine the values of $a, b,$ and $c$.

Step 2: Apply the Condition that Line $L$ is Parallel to the Plane

The line $L$ is parallel to the plane $P: x + 2y - z - 5 = 0$. The direction vector of line $L$ is $\vec{d} = a\hat{i} + b\hat{j} + c\hat{k}$. The normal vector of plane $P$ is $\vec{n} = 1\hat{i} + 2\hat{j} - 1\hat{k}$ (obtained from the coefficients of $x, y, z$).

Since line $L$ is parallel to plane $P$, its direction vector $\vec{d}$ must be perpendicular to the plane's normal vector $\vec{n}$. Therefore, their dot product must be zero: $\vec{d} \cdot \vec{n} = 0$ $(a)(1) + (b)(2) + (c)(-1) = 0$ $a + 2b - c = 0 \quad \text{(Equation 1)}$ This equation provides a crucial relationship between the unknown direction ratios $a, b, c$.

Step 3: Apply the Condition that Line $L$ Intersects Another Line $L_2$

The line $L$ intersects the line $L_2: \frac{x + 1}{-3} = \frac{y - 3}{2} = \frac{z - 2}{-1}$. Let the point of intersection be $B$. We can find the general coordinates of any point on line $L_2$ by setting its symmetric equation equal to a parameter, say $t$: $\frac{x + 1}{-3} = \frac{y - 3}{2} = \frac{z - 2}{-1} = t$ From this, the parametric coordinates of a general point $B$ on $L_2$ are: $x = -1 - 3t$ $y = 3 + 2t$ $z = 2 - t$ So, point $B = (-1 - 3t, 3 + 2t, 2 - t)$.

Since line $L$ passes through point $A(-4, 3, 1)$ and also through point $B$, the direction ratios $(a, b, c)$ of line $L$ must be proportional to the components of the vector $\vec{AB}$. We can write: $a = x_B - x_A = (-1 - 3t) - (-4) = 3 - 3t$ $b = y_B - y_A = (3 + 2t) - 3 = 2t$ $c = z_B - z_A = (2 - t) - 1 = 1 - t$ Now, the direction ratios $a, b, c$ are expressed in terms of the single parameter $t$.

Step 4: Solve for the Parameter $t$

We can substitute the expressions for $a, b, c$ (from Step 3) into Equation 1 (from Step 2): $a + 2b - c = 0$ Substitute: $(3 - 3t) + 2(2t) - (1 - t) = 0$ Expand and simplify the equation: $3 - 3t + 4t - 1 + t = 0$ $2 + 2t = 0$ $2t = -2$ $t = -1$ This value of $t$ identifies the specific point of intersection $B$.

Step 5: Determine the Direction Ratios of Line $L$

Now that we have the value $t = -1$, we can substitute it back into the expressions for $a, b, c$ found in Step 3: $a = 3 - 3t = 3 - 3(-1) = 3 + 3 = 6$ $b = 2t = 2(-1) = -2$ $c = 1 - t = 1 - (-1) = 1 + 1 = 2$ The direction ratios of line $L$ are $(6, -2, 2)$. These can be simplified by dividing by a common factor of 2, resulting in $(3, -1, 1)$.

Step 6: Write the Final Equation of Line $L$

We have the point $A(-4, 3, 1)$ through which line $L$ passes, and its simplified direction ratios $(3, -1, 1)$. Using the symmetric form of the line equation: $\frac{x - (-4)}{3} = \frac{y - 3}{-1} = \frac{z - 1}{1}$ $\frac{x + 4}{3} = \frac{y - 3}{-1} = \frac{z - 1}{1}$

Step 7: Verify with Options

Comparing our derived equation with the given options: (A) $\frac{x + 4}{3} = \frac{y - 3}{-1} = \frac{z - 1}{1}$ (B) $\frac{x + 4}{1} = \frac{y - 3}{1} = \frac{z - 1}{3}$ (C) $\frac{x + 4}{-1} = \frac{y - 3}{1} = \frac{z - 1}{1}$ (D) $\frac{x - 4}{2} = \frac{y + 3}{1} = \frac{z + 1}{4}$

Our equation perfectly matches Option (A).

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Common Mistakes & Tips

• Sign Errors: Pay close attention to negative signs when substituting coordinates or parameter values, especially during subtraction.
• Misinterpreting Parallel Condition: Remember that if a line is parallel to a plane, its direction vector is perpendicular to the plane's normal vector. This leads to their dot product being zero.
• Algebraic Precision: Double-check calculations when expanding and simplifying equations to avoid errors in determining the parameter $t$.
• Simplifying Direction Ratios: While not always mandatory, simplifying direction ratios by dividing by a common factor can make it easier to match your answer with the given options.

Summary

This problem demonstrates a systematic approach to finding the equation of a line using multiple geometric conditions. The strategy involves first setting up the equation of the unknown line with a known point and unknown direction ratios. Then, the condition of being parallel to a plane is used to establish a linear relationship between these direction ratios. Concurrently, the condition of intersecting another line allows us to express these direction ratios in terms of a single parameter. By substituting the parametric expressions into the linear relationship, we solve for the parameter, which in turn determines the precise direction ratios of the line. Finally, these elements are combined to form the complete equation of the line.

The final answer is $\boxed{\text{(A)}}$.