The value of a for which the lines `(x-2)/(1)=(y-9)/(2)=(z-13)/(3) and (x-a)/(-1)=(y-7)/(2)=(z+2)/(-3)` intersect, is

Options

A. `-5`
B. `-2`
C. `5`
D. `-3`

Answer

To find the value of \( a \) for which the lines

\[ \frac{x-2}{1} = \frac{y-9}{2} = \frac{z-13}{3} \]

and

\[ \frac{x-a}{-1} = \frac{y-7}{2} = \frac{z+2}{-3} \]

intersect, we can follow these steps:

### Step 1: Parameterize the first line

Let

\[ \frac{x-2}{1} = \frac{y-9}{2} = \frac{z-13}{3} = t \]

From this, we can express \( x, y, z \) in terms of \( t \):

\[ x = t + 2 \]
\[ y = 2t + 9 \]
\[ z = 3t + 13 \]

### Step 2: Parameterize the second line

Let

\[ \frac{x-a}{-1} = \frac{y-7}{2} = \frac{z+2}{-3} = \lambda \]

From this, we can express \( x, y, z \) in terms of \( \lambda \):

\[ x = -\lambda + a \]
\[ y = 2\lambda + 7 \]
\[ z = -3\lambda - 2 \]

### Step 3: Set the equations equal to each other

Since the lines intersect, we can set the corresponding expressions for \( x, y, z \) equal to each other:

1. \( t + 2 = -\lambda + a \)
2. \( 2t + 9 = 2\lambda + 7 \)
3. \( 3t + 13 = -3\lambda - 2 \)

### Step 4: Solve the equations

From the second equation:

\[ 2t + 9 = 2\lambda + 7 \]

Rearranging gives:

\[ 2t - 2\lambda = -2 \quad \Rightarrow \quad t - \lambda = -1 \quad \Rightarrow \quad \lambda = t + 1 \quad \text{(Equation 1)} \]

From the third equation:

\[ 3t + 13 = -3\lambda - 2 \]

Rearranging gives:

\[ 3t + 15 = -3\lambda \quad \Rightarrow \quad \lambda = -t - 5 \quad \text{(Equation 2)} \]

### Step 5: Equate the two expressions for \( \lambda \)

Setting Equation 1 equal to Equation 2:

\[ t + 1 = -t - 5 \]

Solving for \( t \):

\[ 2t = -6 \quad \Rightarrow \quad t = -3 \]

### Step 6: Substitute \( t \) back to find \( \lambda \)

Using \( t = -3 \) in Equation 1:

\[ \lambda = -3 + 1 = -2 \]

### Step 7: Substitute \( t \) and \( \lambda \) into the first equation

Now substitute \( t = -3 \) into the first equation:

\[ -3 + 2 = -\lambda + a \]

Substituting \( \lambda = -2 \):

\[ -3 + 2 = -(-2) + a \]

This simplifies to:

\[ -1 = 2 + a \]

Solving for \( a \):

\[ a = -1 - 2 = -3 \]

### Conclusion

Thus, the value of \( a \) for which the lines intersect is

\[ \boxed{-3} \]