Key Concepts and Formulas

1. Normal Vector of a Plane: For a plane $Ax+By+Cz+D=0$, the vector $\vec{n} = \langle A, B, C \rangle$ is its normal vector. Any line perpendicular to the plane will have direction ratios proportional to $\langle A, B, C \rangle$.
2. Equation of a Line: A line passing through a point $(x_1, y_1, z_1)$ with direction ratios $\langle a, b, c \rangle$ can be expressed in its parametric form as: $x = x_1 + a\lambda, \quad y = y_1 + b\lambda, \quad z = z_1 + c\lambda$ where $\lambda$ is a scalar parameter.
3. Mirror Image of a Point with Respect to a Plane: The coordinates of the mirror image $Q(x', y', z')$ of a point $P(x_1, y_1, z_1)$ with respect to a plane $S: Ax+By+Cz+D=0$ are given by the formula: $\frac{x' - x_1}{A} = \frac{y' - y_1}{B} = \frac{z' - z_1}{C} = \frac{-2(Ax_1+By_1+Cz_1+D)}{A^2+B^2+C^2}$
4. Distance Between Two Points: The square of the distance $d^2$ between two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ is given by: $d^2 = (x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2$

---

Step-by-Step Solution

1. Determine the Direction Ratios of Line PQ

• Understanding the relationship: Point $Q$ is the mirror image of point $P$ with respect to the plane $S$. This means that the line segment $PQ$ is perpendicular (normal) to the plane $S$.
• Extracting information from the plane: The given plane is $S: x + y + z = 5$, which can be rewritten as $x + y + z - 5 = 0$. Comparing this to the general plane equation $Ax+By+Cz+D=0$, we identify the coefficients $A=1, B=1, C=1$.
• Applying the concept: The direction ratios of the normal to the plane $S$ are $\langle A, B, C \rangle = \langle 1, 1, 1 \rangle$. Since line $PQ$ is normal to the plane, its direction ratios are also $\langle 1, 1, 1 \rangle$.
• Result: The direction ratios of line $PQ$ are $\langle 1, 1, 1 \rangle$.

2. Formulate the Equation of Line L

• Understanding the given information: We are told that line $L$ passes through the point $(1, -1, -1)$ and is parallel to the line $PQ$.
• Applying the concept of parallel lines: Parallel lines share the same direction ratios. From Step 1, the direction ratios of $PQ$ are $\langle 1, 1, 1 \rangle$. Therefore, the direction ratios of line $L$ are also $\langle 1, 1, 1 \rangle$.
• Constructing the line equation: Using the point $(x_1, y_1, z_1) = (1, -1, -1)$ and direction ratios $\langle a, b, c \rangle = \langle 1, 1, 1 \rangle$, we can write the parametric form of line $L$: $x = 1 + 1\lambda \Rightarrow x = 1 + \lambda$ $y = -1 + 1\lambda \Rightarrow y = -1 + \lambda$ $z = -1 + 1\lambda \Rightarrow z = -1 + \lambda$
• Result: Any point on line $L$ can be represented as $(1+\lambda, -1+\lambda, -1+\lambda)$.

3. Find the Coordinates of Point R

• Understanding the definition of R: Point $R$ is the intersection of line $L$ and plane $S$. This means that the coordinates of $R$ must satisfy both the equation of line $L$ and the equation of plane $S$.
• Substituting line into plane: We substitute the parametric coordinates of a general point on line $L$ (from Step 2) into the equation of plane $S: x + y + z = 5$. $(1+\lambda) + (-1+\lambda) + (-1+\lambda) = 5$
• Solving for the parameter $\lambda$: $3\lambda - 1 = 5$ $3\lambda = 6$ $\lambda = 2$
• Finding the coordinates of R: Substitute $\lambda = 2$ back into the parametric equations of line $L$: $x_R = 1 + 2 = 3$ $y_R = -1 + 2 = 1$ $z_R = -1 + 2 = 1$
• Result: The coordinates of point $R$ are $(3, 1, 1)$.

4. Find the Coordinates of Point Q

• Understanding the definition of Q: Point $Q$ is the mirror image of $P(1, 0, 1)$ with respect to the plane $S: x + y + z - 5 = 0$.
• Identifying parameters for the mirror image formula:
  • Point $P(x_1, y_1, z_1) = (1, 0, 1)$
  • Plane $S: Ax+By+Cz+D=0$, so $A=1, B=1, C=1, D=-5$.
• Calculating the numerator term: $Ax_1+By_1+Cz_1+D = (1)(1) + (1)(0) + (1)(1) + (-5) = 1 + 0 + 1 - 5 = -3$
• Calculating the denominator term: $A^2+B^2+C^2 = 1^2 + 1^2 + 1^2 = 1 + 1 + 1 = 3$
• Applying the mirror image formula: Let $Q(x', y', z')$ be the image. $\frac{x' - 1}{1} = \frac{y' - 0}{1} = \frac{z' - 1}{1} = \frac{-2(-3)}{3}$ $\frac{x' - 1}{1} = \frac{y'}{1} = \frac{z' - 1}{1} = \frac{6}{3}$ $\frac{x' - 1}{1} = \frac{y'}{1} = \frac{z' - 1}{1} = 2$
• Solving for $x', y', z'$: $x' - 1 = 2 \Rightarrow x' = 3$ $y' = 2$ $z' - 1 = 2 \Rightarrow z' = 3$
• Result: The coordinates of point $Q$ are $(3, 2, 3)$.

**5. Calculate $QR^2$}

• Understanding the final objective: We need to find the square of the distance between point $Q(3, 2, 3)$ and point $R(3, 1, 1)$.
• Applying the distance formula: $QR^2 = (x_R - x_Q)^2 + (y_R - y_Q)^2 + (z_R - z_Q)^2$ $QR^2 = (3 - 3)^2 + (1 - 2)^2 + (1 - 3)^2$ $QR^2 = (0)^2 + (-1)^2 + (-2)^2$ $QR^2 = 0 + 1 + 4$ $QR^2 = 5$
• Final Answer: The value of $QR^2$ is 5.

---

Tips and Common Mistakes to Avoid

• Direction Ratios for Perpendicularity: Always remember that the line connecting a point and its mirror image with respect to a plane is perpendicular to the plane. Therefore, its direction ratios are directly given by the coefficients of $x, y, z$ in the plane's equation (the normal vector).
• Sign Errors in Mirror Image Formula: The mirror image formula includes a $-2$ multiplier in the numerator. Double-check the sign of $Ax_1+By_1+Cz_1+D$ and ensure the $-2$ is correctly applied. A common mistake is to use -1 instead of -2, which would yield the foot of the perpendicular instead of the mirror image.
• Parametric Form for Intersection: When finding the intersection of a line and a plane, using the parametric form of the line is generally the most straightforward method. Substitute the parametric expressions for $x, y, z$ into the plane equation to find the parameter value.

Summary and Key Takeaway

This problem is a comprehensive test of 3D geometry fundamentals. We systematically determined the direction of a line perpendicular to a plane, formulated the equation of a parallel line, found the intersection of a line and a plane, and accurately calculated the mirror image of a point. The final calculation of $QR^2$ showed that the square of the distance between the two points is 5. The key takeaway is to methodically break down complex problems into smaller, manageable steps, applying the correct formula or concept at each stage.

The final answer is $\boxed{5}$ which corresponds to option (B).