Key Concepts and Formulas

1. Condition for a Point to Lie on a Plane: A point $P(x_0, y_0, z_0)$ lies on a plane $Ax + By + Cz + D = 0$ if and only if its coordinates satisfy the plane's equation: $Ax_0 + By_0 + Cz_0 + D = 0$.
2. Mirror Image of a Point with Respect to a Plane: If $S(x_2, y_2, z_2)$ is the mirror image of a point $Q(x_1, y_1, z_1)$ with respect to a plane $Ax + By + Cz + D = 0$, then:
   • The line segment $QS$ is perpendicular to the plane. The direction ratios (DRs) of the line $QS$ are $(A, B, C)$.
   • The foot of the perpendicular, $F$, from $Q$ to the plane is the midpoint of the line segment $QS$. The coordinates of $S$ can be found using the formula: $\frac{x_2 - x_1}{A} = \frac{y_2 - y_1}{B} = \frac{z_2 - z_1}{C} = -2 \frac{Ax_1 + By_1 + Cz_1 + D}{A^2 + B^2 + C^2}$ Alternatively, one can find the foot of the perpendicular $F$ first, and then use the midpoint formula.
3. Distance Formula in 3D: The square of the distance between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is $d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2$.

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Step-by-Step Solution

1. Determine the unknown coordinate $\gamma$ for point R.

• Explanation: We are given that point $R(3, 5, \gamma)$ lies on the plane $2x - y + z + 3 = 0$. For a point to lie on a plane, its coordinates must satisfy the plane's equation.
• Working: Substitute the coordinates of $R(3, 5, \gamma)$ into the plane's equation: $2(3) - (5) + (\gamma) + 3 = 0$ $6 - 5 + \gamma + 3 = 0$ $1 + \gamma + 3 = 0$ $\gamma + 4 = 0$ $\gamma = -4$
• Result: The coordinates of point $R$ are $(3, 5, -4)$.

2. Determine the coordinates of the mirror image S.

• Explanation: We need to find the mirror image $S(x_S, y_S, z_S)$ of point $Q(1, 3, 4)$ with respect to the plane $2x - y + z + 3 = 0$. We will use the formula for the mirror image directly. The plane equation is $Ax + By + Cz + D = 0$, where $A=2, B=-1, C=1, D=3$. The point is $Q(x_Q, y_Q, z_Q) = (1, 3, 4)$.
• Working: Apply the mirror image formula: $\frac{x_S - x_Q}{A} = \frac{y_S - y_Q}{B} = \frac{z_S - z_Q}{C} = -2 \frac{Ax_Q + By_Q + Cz_Q + D}{A^2 + B^2 + C^2}$ First, calculate the numerator and denominator of the fraction: $Ax_Q + By_Q + Cz_Q + D = 2(1) - 1(3) + 1(4) + 3 = 2 - 3 + 4 + 3 = 6$ $A^2 + B^2 + C^2 = 2^2 + (-1)^2 + 1^2 = 4 + 1 + 1 = 6$ Now substitute these values into the formula: $\frac{x_S - 1}{2} = \frac{y_S - 3}{-1} = \frac{z_S - 4}{1} = -2 \frac{6}{6}$ $\frac{x_S - 1}{2} = \frac{y_S - 3}{-1} = \frac{z_S - 4}{1} = -2$ Now, solve for $x_S, y_S, z_S$: $x_S - 1 = 2(-2) \Rightarrow x_S - 1 = -4 \Rightarrow x_S = -3$ $y_S - 3 = -1(-2) \Rightarrow y_S - 3 = 2 \Rightarrow y_S = 5$ $z_S - 4 = 1(-2) \Rightarrow z_S - 4 = -2 \Rightarrow z_S = 2$
• Result: The coordinates of the mirror image $S$ are $(-3, 5, 2)$.

3. Calculate the square of the length of the line segment SR.

• Explanation: We need to find the square of the distance between point $S(-3, 5, 2)$ and point $R(3, 5, -4)$. We use the 3D distance formula.
• Working: $SR^2 = (x_R - x_S)^2 + (y_R - y_S)^2 + (z_R - z_S)^2$ $SR^2 = (3 - (-3))^2 + (5 - 5)^2 + (-4 - 2)^2$ $SR^2 = (3 + 3)^2 + (0)^2 + (-6)^2$ $SR^2 = (6)^2 + 0 + 36$ $SR^2 = 36 + 0 + 36$ $SR^2 = 72$
• Result: The square of the length of the line segment $SR$ is $72$.

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Common Mistakes & Tips

• Sign Errors: Be extremely careful with negative signs, especially when substituting coordinates or calculating differences.
• Formula for Mirror Image: Remember the factor of $-2$ in the mirror image formula. If finding the foot of the perpendicular, it's just $-1$.
• Direction Ratios vs. Normal Vector: The coefficients of $x, y, z$ in the plane equation directly give the direction ratios of the normal vector, which are used for the perpendicular line.
• Double-Check Calculations: Simple arithmetic errors are common in 3D geometry problems due to multiple steps and coordinates.

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Summary

This problem effectively tests the understanding of points, planes, and distances in 3D geometry. We first determined the unknown coordinate of point R by using the condition that it lies on the plane. Next, we found the coordinates of the mirror image S using the dedicated formula, which involves calculating the value of the expression $Ax_1 + By_1 + Cz_1 + D$ and the sum of squares of direction ratios. Finally, we applied the 3D distance formula to calculate the square of the length of the line segment SR. The derived square of the length of SR is 72.

The final answer is $\boxed{3}$.