The vector equations of two lines `L_1 and L_2` are respectively `vec r=17hat i–9hat j+9hat k+ lambda(3hat i +hat j+ 5hat k) and vec r=15hat –8hat j-hat k+mu(4hat i+3hat j)` `I` `L_1 and L_2` are skew lines `II` `(11, -11, -1)` is the point of intersection of `L_1 and L_2` `III` `(-11, 11, 1)` is the point of intersection of `L_1 and L_2`. `IV ` `cos^-1 (3/sqrt35)` is the acute angle between `_1 and L_2` then , Which of the following is true?

Options

A. II and IV
B. I and IV
C. Only IV
D. III and IV

Answer

To solve the problem, we need to analyze the vector equations of the two lines \( L_1 \) and \( L_2 \) and evaluate the given statements.

### Step 1: Write down the vector equations
The vector equations of the lines are given as:
- \( L_1: \vec{r} = 17\hat{i} - 9\hat{j} + 9\hat{k} + \lambda(3\hat{i} + \hat{j} + 5\hat{k}) \)
- \( L_2: \vec{r} = 15\hat{i} - 8\hat{j} - \hat{k} + \mu(4\hat{i} + 3\hat{j}) \)

### Step 2: Identify direction vectors and points
From the equations:
- For \( L_1 \), the direction vector \( \vec{d_1} = 3\hat{i} + \hat{j} + 5\hat{k} \) and a point on the line is \( P_1(17, -9, 9) \).
- For \( L_2 \), the direction vector \( \vec{d_2} = 4\hat{i} + 3\hat{j} \) and a point on the line is \( P_2(15, -8, -1) \).

### Step 3: Check if the lines are skew
To determine if the lines are skew, we need to check if they are parallel and if they intersect.

1. **Check for parallelism**:
- The lines are parallel if \( \vec{d_1} \) is a scalar multiple of \( \vec{d_2} \).
- \( \vec{d_1} = (3, 1, 5) \) and \( \vec{d_2} = (4, 3, 0) \).
- The ratios \( \frac{3}{4}, \frac{1}{3}, \frac{5}{0} \) are not equal, hence the lines are not parallel.

2. **Check for intersection**:
- Set the equations equal to each other:
\[ 17 + 3\lambda = 15 + 4\mu \quad (1) \]
\[ -9 + \lambda = -8 + 3\mu \quad (2) \]
\[ 9 + 5\lambda = -1 \quad (3) \]
- From equation (3), \( 5\lambda = -10 \Rightarrow \lambda = -2 \).
- Substitute \( \lambda = -2 \) in equation (1):
\[ 17 + 3(-2) = 15 + 4\mu \Rightarrow 11 = 15 + 4\mu \Rightarrow 4\mu = -4 \Rightarrow \mu = -1. \]
- Substitute \( \lambda = -2 \) in equation (2):
\[ -9 + (-2) = -8 + 3(-1) \Rightarrow -11 = -11. \]
- Since both equations are satisfied, the lines intersect at the point \( (11, -11, -1) \).

### Step 4: Evaluate the statements
1. **Statement I**: \( L_1 \) and \( L_2 \) are skew lines. **False** (they intersect).
2. **Statement II**: \( (11, -11, -1) \) is the point of intersection of \( L_1 \) and \( L_2 \). **True**.
3. **Statement III**: \( (-11, 11, 1) \) is the point of intersection of \( L_1 \) and \( L_2 \). **False** (not the intersection point).
4. **Statement IV**: \( \cos^{-1} \left( \frac{3}{\sqrt{35}} \right) \) is the acute angle between \( L_1 \) and \( L_2 \). **True** (we will verify this).

### Step 5: Calculate the angle between the lines
To find the angle \( \theta \) between the lines:
- Use the formula:
\[ \cos \theta = \frac{\vec{d_1} \cdot \vec{d_2}}{|\vec{d_1}| |\vec{d_2}|} \]
- Calculate \( \vec{d_1} \cdot \vec{d_2} \):
\[ \vec{d_1} \cdot \vec{d_2} = 3 \cdot 4 + 1 \cdot 3 + 5 \cdot 0 = 12 + 3 + 0 = 15. \]
- Calculate magnitudes:
\[ |\vec{d_1}| = \sqrt{3^2 + 1^2 + 5^2} = \sqrt{9 + 1 + 25} = \sqrt{35}, \]
\[ |\vec{d_2}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5. \]
- Substitute into the cosine formula:
\[ \cos \theta = \frac{15}{\sqrt{35} \cdot 5} = \frac{3}{\sqrt{35}}. \]
- Therefore, \( \theta = \cos^{-1} \left( \frac{3}{\sqrt{35}} \right) \).

### Conclusion
The true statements are:
- Statement II and Statement IV are true.