Find the angle between the lines `(x-2)/(3)=(y-3)/(6)=(z-4)/(8)and(x+2)/(2)=(y+3)/(5)=(z+4)/(9)`.

Options

A. `cos^(-1)((108)/(sqrt11990))`
B. `cos^(-1)((99)/(sqrt9701))`
C. `cos^(-1)((9)/(sqrt9701))`
D. `cos^(-1)((10)/(sqrt9701))`

Answer

To find the angle between the two lines given in the question, we will follow these steps:

### Step 1: Identify the direction ratios of the lines

The lines are given in the symmetric form:

1. For the first line: \((x-2)/(3)=(y-3)/(6)=(z-4)/(8)\)
- The direction ratios are \(a_1 = 3\), \(b_1 = 6\), \(c_1 = 8\).

2. For the second line: \((x+2)/(2)=(y+3)/(5)=(z+4)/(9)\)
- The direction ratios are \(a_2 = 2\), \(b_2 = 5\), \(c_2 = 9\).

### Step 2: Use the formula for the cosine of the angle between two lines

The formula to find the cosine of the angle \(\theta\) between two lines with direction ratios \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\) is:

\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}} \]

### Step 3: Calculate the numerator

Now, we will calculate the numerator \(a_1 a_2 + b_1 b_2 + c_1 c_2\):

\[ = (3 \cdot 2) + (6 \cdot 5) + (8 \cdot 9) \]
\[ = 6 + 30 + 72 = 108 \]

### Step 4: Calculate the denominator

Next, we will calculate the denominator:

1. Calculate \(\sqrt{a_1^2 + b_1^2 + c_1^2}\):

\[ \sqrt{3^2 + 6^2 + 8^2} = \sqrt{9 + 36 + 64} = \sqrt{109} \]

2. Calculate \(\sqrt{a_2^2 + b_2^2 + c_2^2}\):

\[ \sqrt{2^2 + 5^2 + 9^2} = \sqrt{4 + 25 + 81} = \sqrt{110} \]

3. Now, the denominator is:

\[ \sqrt{109} \cdot \sqrt{110} = \sqrt{109 \cdot 110} \]

### Step 5: Substitute values into the cosine formula

Now we can substitute the values into the formula:

\[ \cos \theta = \frac{108}{\sqrt{109 \cdot 110}} \]

### Step 6: Calculate \(\theta\)

To find \(\theta\), we take the inverse cosine:

\[ \theta = \cos^{-1}\left(\frac{108}{\sqrt{109 \cdot 110}}\right) \]

### Final Result

This gives us the angle between the two lines.

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