1. Key Concepts and Formulas

• Mirror Image Property: If a point $P(x_1, y_1, z_1)$ has a mirror image $Q(x_2, y_2, z_2)$ with respect to a line $L$, then the midpoint $M$ of the line segment $PQ$ must lie on the line $L$. Additionally, the line segment $PQ$ is perpendicular to the line $L$. For this problem, the midpoint property alone is sufficient to solve for the unknowns.
• Midpoint Formula: The coordinates of the midpoint $M$ of a line segment connecting $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are given by: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)$
• Symmetric Form of a Line: A line passing through a point $(x_0, y_0, z_0)$ with direction ratios $(d_x, d_y, d_z)$ can be represented as: $\frac{x - x_0}{d_x} = \frac{y - y_0}{d_y} = \frac{z - z_0}{d_z}$

2. Step-by-Step Solution

Step 1: Identify Given Information and Key Geometric Property

We are given:

• The original point $P(a, 6, 9)$.
• Its mirror image $Q(20, b, -a-9)$.
• The line of reflection $L: \frac{x - 3}{7} = \frac{y - 2}{5} = \frac{z - 1}{-9}$.

The fundamental property we will use is that the midpoint of the line segment $PQ$ must lie on the line $L$.

Step 2: Calculate the Coordinates of the Midpoint M of PQ

Using the midpoint formula for $P(a, 6, 9)$ and $Q(20, b, -a-9)$: $M = \left( \frac{a + 20}{2}, \frac{6 + b}{2}, \frac{9 + (-a - 9)}{2} \right)$ Simplify the coordinates of $M$: $M = \left( \frac{a + 20}{2}, \frac{6 + b}{2}, \frac{-a}{2} \right)$

Step 3: Substitute Midpoint M into the Line Equation

Since the midpoint $M$ lies on the line $L$, its coordinates must satisfy the line's equation. Substitute the coordinates of $M$ into the symmetric equation of line $L$: $\frac{\left(\frac{a + 20}{2}\right) - 3}{7} = \frac{\left(\frac{6 + b}{2}\right) - 2}{5} = \frac{\left(\frac{-a}{2}\right) - 1}{-9}$ Now, simplify the numerators:

• First numerator: $\frac{a + 20}{2} - 3 = \frac{a + 20 - 6}{2} = \frac{a + 14}{2}$
• Second numerator: $\frac{6 + b}{2} - 2 = \frac{6 + b - 4}{2} = \frac{b + 2}{2}$
• Third numerator: $\frac{-a}{2} - 1 = \frac{-a - 2}{2}$

Substitute these simplified numerators back into the equation: $\frac{\frac{a + 14}{2}}{7} = \frac{\frac{b + 2}{2}}{5} = \frac{\frac{-a - 2}{2}}{-9}$ This simplifies to: $\frac{a + 14}{14} = \frac{b + 2}{10} = \frac{-(a + 2)}{-18}$ Further simplifying the third term by canceling the negative signs: $\frac{a + 14}{14} = \frac{b + 2}{10} = \frac{a + 2}{18}$

Step 4: Form and Solve Equations for 'a'

We equate the first and third ratios to solve for $a$: $\frac{a + 14}{14} = \frac{a + 2}{18}$ Cross-multiply to eliminate the denominators: $18(a + 14) = 14(a + 2)$ $18a + 252 = 14a + 28$ Collect terms with $a$ on one side and constants on the other: $18a - 14a = 28 - 252$ $4a = -224$ Divide by 4: $a = -56$

Step 5: Form and Solve Equations for 'b'

Now, we use the second and third ratios, along with the value of $a = -56$, to solve for $b$: $\frac{b + 2}{10} = \frac{a + 2}{18}$ Substitute $a = -56$: $\frac{b + 2}{10} = \frac{-56 + 2}{18}$ $\frac{b + 2}{10} = \frac{-54}{18}$ Simplify the right side: $\frac{b + 2}{10} = -3$ Multiply both sides by 10: $b + 2 = -30$ Subtract 2 from both sides: $b = -30 - 2$ $b = -32$

Step 6: Calculate the Required Value |a + b|

The problem asks for the value of $|a + b|$. Substitute the values $a = -56$ and $b = -32$: $|a + b| = |-56 + (-32)|$ $|a + b| = |-56 - 32|$ $|a + b| = |-88|$ The absolute value of $-88$ is $88$: $|a + b| = 88$

3. Common Mistakes & Tips

• Sign Errors: Be vigilant with negative signs, especially when simplifying expressions like $\frac{-a-2}{-18}$ or during algebraic manipulations involving subtraction and addition of negative numbers.
• Fraction Simplification: Take care when simplifying complex fractions, ensure you multiply the denominator of the inner fraction correctly with the outer denominator. For example, $\frac{(A/B)}{C} = \frac{A}{BC}$.
• Algebraic Precision: Double-check your arithmetic when solving linear equations. A small calculation error can propagate and lead to an incorrect final answer.

4. Summary

This problem leveraged the fundamental property of a mirror image in 3D geometry: the midpoint of the original point and its image must lie on the line of reflection. By calculating the midpoint of the given points $P$ and $Q$ (which contained unknown variables $a$ and $b$), and then substituting these midpoint coordinates into the symmetric equation of the line, we formed a system of equations. Solving this system allowed us to determine the values of $a = -56$ and $b = -32$. Finally, we computed the required value $|a+b|$, which was found to be 88.

5. Final Answer

The final answer is $\boxed{88}$, which corresponds to option (A).