If the angle between the lines, `x/2=y/2=z/1 and (5-x)/(-2) = (7y-14)/(p ) = (z-3)/(4) is cos ^(-1) ((2)/(3)),` then P is equal to

Options

A. `-4/7`
B. `7/2`
C. `-7/4`
D. `7/2`

Answer

To solve the problem, we need to find the value of \( p \) given that the angle between the two lines is \( \cos^{-1} \left( \frac{2}{3} \right) \).

### Step 1: Find the direction ratios of the first line
The first line is given by the equation:
\[ \frac{x}{2} = \frac{y}{2} = \frac{z}{1} \]
From this, we can deduce the direction ratios (or direction cosines) of the first line:
\[ l_1 = 2, \quad m_1 = 2, \quad n_1 = 1 \]

### Step 2: Find the direction ratios of the second line
The second line is given by the equation:
\[ \frac{5 - x}{-2} = \frac{7y - 14}{p} = \frac{z - 3}{4} \]
Rearranging the first part:
\[ \frac{x - 5}{2} = \frac{7y - 14}{p} = \frac{z - 3}{4} \]
This gives us the direction ratios:
\[ l_2 = 2, \quad m_2 = \frac{p}{7}, \quad n_2 = 4 \]

### Step 3: Use the formula for the cosine of the angle between two lines
The cosine of the angle \( \theta \) between two lines can be calculated using the formula:
\[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \sqrt{l_2^2 + m_2^2 + n_2^2}} \]
Given that \( \cos \theta = \frac{2}{3} \), we can set up the equation:
\[ \frac{2 \cdot 2 + 2 \cdot \frac{p}{7} + 1 \cdot 4}{\sqrt{2^2 + 2^2 + 1^2} \sqrt{2^2 + \left(\frac{p}{7}\right)^2 + 4^2}} = \frac{2}{3} \]

### Step 4: Calculate the magnitudes
Calculating the magnitudes:
\[ \sqrt{l_1^2 + m_1^2 + n_1^2} = \sqrt{2^2 + 2^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]
\[ \sqrt{l_2^2 + m_2^2 + n_2^2} = \sqrt{2^2 + \left(\frac{p}{7}\right)^2 + 4^2} = \sqrt{4 + \frac{p^2}{49} + 16} = \sqrt{20 + \frac{p^2}{49}} \]

### Step 5: Substitute and simplify
Substituting the values back into the cosine formula:
\[ \frac{4 + \frac{2p}{7} + 4}{3 \sqrt{20 + \frac{p^2}{49}}} = \frac{2}{3} \]
This simplifies to:
\[ \frac{8 + \frac{2p}{7}}{3 \sqrt{20 + \frac{p^2}{49}}} = \frac{2}{3} \]

### Step 6: Cross-multiply and simplify further
Cross-multiplying gives:
\[ 8 + \frac{2p}{7} = 2 \sqrt{20 + \frac{p^2}{49}} \]
Squaring both sides:
\[ (8 + \frac{2p}{7})^2 = 4(20 + \frac{p^2}{49}) \]

### Step 7: Expand and solve for \( p \)
Expanding both sides:
\[ 64 + \frac{32p}{7} + \frac{4p^2}{49} = 80 + \frac{4p^2}{49} \]
Cancelling \( \frac{4p^2}{49} \) from both sides:
\[ 64 + \frac{32p}{7} = 80 \]
Rearranging gives:
\[ \frac{32p}{7} = 16 \]
Multiplying both sides by \( 7 \):
\[ 32p = 112 \]
Thus:
\[ p = \frac{112}{32} = \frac{7}{2} \]

### Final Answer
The value of \( p \) is:
\[ \boxed{\frac{7}{2}} \]