Let `L_(1):r=(i+5j+5k)+t(4i-4j+5k)" and "L_(2): r=(2i+4j+5k)+t(8i-3j+k)` be two lines then

Options

A. `L_(1)` is parallel to `L_(2)`
B. `L_(1)` is parallel to `L_(2)`
C. `L_(1)` is not parallel to `L_(2)`
D. none of these

Answer

To determine whether the two lines \( L_1 \) and \( L_2 \) are parallel, we need to analyze their direction ratios.

### Given Lines:
1. **Line \( L_1 \)**:
\[ \mathbf{r} = (i + 5j + 5k) + t(4i - 4j + 5k) \]
- Position vector: \( \mathbf{a_1} = i + 5j + 5k \)
- Direction vector: \( \mathbf{d_1} = 4i - 4j + 5k \)

2. **Line \( L_2 \)**:
\[ \mathbf{r} = (2i + 4j + 5k) + t(8i - 3j + k) \]
- Position vector: \( \mathbf{a_2} = 2i + 4j + 5k \)
- Direction vector: \( \mathbf{d_2} = 8i - 3j + k \)

### Step 1: Identify Direction Ratios
- For \( L_1 \), the direction ratios are \( (4, -4, 5) \).
- For \( L_2 \), the direction ratios are \( (8, -3, 1) \).

### Step 2: Check for Scalar Multiples
To check if the lines are parallel, we need to see if the direction ratios are scalar multiples of each other. This means we need to find a scalar \( \lambda \) such that:
\[ (4, -4, 5) = \lambda (8, -3, 1) \]

### Step 3: Set Up Equations
From the above, we can set up the following equations based on the components:
1. \( 4 = 8\lambda \)
2. \( -4 = -3\lambda \)
3. \( 5 = 1\lambda \)

### Step 4: Solve for \( \lambda \)
1. From \( 4 = 8\lambda \):
\[ \lambda = \frac{4}{8} = \frac{1}{2} \]

2. From \( -4 = -3\lambda \):
\[ \lambda = \frac{-4}{-3} = \frac{4}{3} \]

3. From \( 5 = 1\lambda \):
\[ \lambda = 5 \]

### Step 5: Compare Values of \( \lambda \)
The values of \( \lambda \) obtained from the three equations are:
- From the first equation: \( \lambda = \frac{1}{2} \)
- From the second equation: \( \lambda = \frac{4}{3} \)
- From the third equation: \( \lambda = 5 \)

Since all three values of \( \lambda \) are different, the direction ratios are not scalar multiples of each other.

### Conclusion
Since the direction ratios of \( L_1 \) and \( L_2 \) are not scalar multiples of each other, we conclude that:
\[ \text{Line } L_1 \text{ is not parallel to Line } L_2. \]

Thus, the correct option is:
**C: \( L_1 \) is not parallel to \( L_2 \)**.

---