STATEMENT-1 : The shortest distance between the skew lines `(x+3)/-4=(y-6)/3=\z/2 and (x+2)/-4=y/1=(z-7)/1` is 9. and STATEMENT-2 : Two lines are skew lines if there doesn't exist a plane passing through them.

Options

A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement -1
B. Statement-1 is True, Statement-2 is true, Statement-2 is NOT a correct explanation for Statement -1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we need to analyze the two statements provided regarding the skew lines and determine their validity.

### Step 1: Identify the lines from the given equations
The equations of the lines are given in the symmetric form:
1. Line 1: \((x + 3)/-4 = (y - 6)/3 = z/2\)
2. Line 2: \((x + 2)/-4 = y/1 = (z - 7)/1\)

### Step 2: Convert the equations into vector form
From the symmetric equations, we can express the lines in vector form:
- For Line 1:
- Point on Line 1 (A1): \((-3, 6, 0)\)
- Direction vector (B1): \((-4, 3, 2)\)

- For Line 2:
- Point on Line 2 (A2): \((-2, 0, 7)\)
- Direction vector (B2): \((-4, 1, 1)\)

### Step 3: Calculate the cross product of the direction vectors
To find the shortest distance between the two skew lines, we first need to calculate the cross product of the direction vectors B1 and B2.

\[ B1 \times B2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -4 & 3 & 2 \\ -4 & 1 & 1 \end{vmatrix} \]

Calculating the determinant:
\[ B1 \times B2 = \hat{i} \begin{vmatrix} 3 & 2 \\ 1 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} -4 & 2 \\ -4 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} -4 & 3 \\ -4 & 1 \end{vmatrix} \]
\[ = \hat{i} (3 \cdot 1 - 2 \cdot 1) - \hat{j} (-4 \cdot 1 - (-4 \cdot 2)) + \hat{k} (-4 \cdot 1 - (-4 \cdot 3)) \]
\[ = \hat{i} (3 - 2) - \hat{j} (-4 + 8) + \hat{k} (-4 + 12) \]
\[ = \hat{i} (1) - \hat{j} (4) + \hat{k} (8) \]
Thus,
\[ B1 \times B2 = (1, -4, 8) \]

### Step 4: Calculate the magnitude of the cross product
The magnitude of the cross product is given by:
\[ |B1 \times B2| = \sqrt{1^2 + (-4)^2 + 8^2} = \sqrt{1 + 16 + 64} = \sqrt{81} = 9 \]

### Step 5: Find the vector from A1 to A2
Now, we need to find the vector \(A2 - A1\):
\[ A2 - A1 = (-2, 0, 7) - (-3, 6, 0) = (1, -6, 7) \]

### Step 6: Calculate the dot product
Next, we calculate the dot product of the cross product and the vector \(A2 - A1\):
\[ (B1 \times B2) \cdot (A2 - A1) = (1, -4, 8) \cdot (1, -6, 7) = 1 \cdot 1 + (-4) \cdot (-6) + 8 \cdot 7 \]
\[ = 1 + 24 + 56 = 81 \]

### Step 7: Calculate the shortest distance
Finally, the shortest distance \(d\) between the two skew lines is given by:
\[ d = \frac{|(B1 \times B2) \cdot (A2 - A1)|}{|B1 \times B2|} = \frac{81}{9} = 9 \]

### Conclusion
- **Statement 1** is true: The shortest distance between the skew lines is indeed 9.
- **Statement 2** is also true: Two lines are skew lines if there does not exist a plane passing through them.

However, Statement 2 does not explain Statement 1, as the shortest distance calculation is independent of the definition of skew lines.

### Final Answer
Statement 1 is true, Statement 2 is true, but Statement 2 is not a correct explanation for Statement 1.

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