Let `a, b in R`. If the mirror image of the point P(a, 6, 9) with respect to the line `(x-3)/(7)=(y-2)/(5)=(z-1)/(-9)` is (20, b, -a -9), then |a + b| is equal to :

Options

A. 88
B. 86
C. 84
D. 90

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that the mirror image of the point \( P(a, 6, 9) \) with respect to the line given by the equations \( \frac{x-3}{7} = \frac{y-2}{5} = \frac{z-1}{-9} \) is \( (20, b, -a - 9) \).

### Step 1: Identify the midpoint of the points P and Q
The midpoint \( M \) of points \( P(a, 6, 9) \) and \( Q(20, b, -a - 9) \) can be calculated using the midpoint formula:
\[ M = \left( \frac{a + 20}{2}, \frac{6 + b}{2}, \frac{9 + (-a - 9)}{2} \right) = \left( \frac{a + 20}{2}, \frac{6 + b}{2}, \frac{-a}{2} \right) \]

### Step 2: Substitute the midpoint into the line equation
The line is represented in symmetric form as:
\[ \frac{x-3}{7} = \frac{y-2}{5} = \frac{z-1}{-9} \]
Let \( t \) be the parameter such that:
\[ x = 7t + 3, \quad y = 5t + 2, \quad z = -9t + 1 \]
To find the value of \( t \) that corresponds to the midpoint \( M \), we set:
\[ \frac{\frac{a + 20}{2} - 3}{7} = \frac{\frac{6 + b}{2} - 2}{5} = \frac{\frac{-a}{2} - 1}{-9} \]

### Step 3: Equate the first and second expressions
Starting with the first two expressions:
\[ \frac{\frac{a + 20}{2} - 3}{7} = \frac{\frac{6 + b}{2} - 2}{5} \]
Cross-multiplying gives:
\[ 5\left(\frac{a + 20}{2} - 3\right) = 7\left(\frac{6 + b}{2} - 2\right) \]
Expanding both sides:
\[ 5\left(\frac{a + 20 - 6}{2}\right) = 7\left(\frac{b - 2}{2}\right) \]
\[ \frac{5(a + 14)}{2} = \frac{7(b - 2)}{2} \]
Multiplying through by 2:
\[ 5(a + 14) = 7(b - 2) \]
Expanding:
\[ 5a + 70 = 7b - 14 \]
Rearranging gives:
\[ 5a - 7b = -84 \quad \text{(Equation 1)} \]

### Step 4: Equate the first and third expressions
Now, equate the first and third expressions:
\[ \frac{\frac{a + 20}{2} - 3}{7} = \frac{\frac{-a}{2} - 1}{-9} \]
Cross-multiplying gives:
\[ -9\left(\frac{a + 20}{2} - 3\right) = 7\left(\frac{-a}{2} - 1\right) \]
Expanding both sides:
\[ -9\left(\frac{a + 20 - 6}{2}\right) = 7\left(\frac{-a - 2}{2}\right) \]
\[ -\frac{9(a + 14)}{2} = \frac{-7(a + 2)}{2} \]
Multiplying through by 2:
\[ -9(a + 14) = -7(a + 2) \]
Expanding:
\[ -9a - 126 = -7a - 14 \]
Rearranging gives:
\[ -2a = 112 \quad \Rightarrow \quad a = -56 \quad \text{(Equation 2)} \]

### Step 5: Substitute \( a \) back into Equation 1
Substituting \( a = -56 \) into Equation 1:
\[ 5(-56) - 7b = -84 \]
\[ -280 - 7b = -84 \]
\[ -7b = 196 \quad \Rightarrow \quad b = -28 \]

### Step 6: Calculate \( |a + b| \)
Now we calculate \( |a + b| \):
\[ |a + b| = |-56 - 28| = |-84| = 84 \]

### Final Answer
Thus, the value of \( |a + b| \) is \( \boxed{84} \).