Find the value of so that the lines
`(-(x-1))/(3)=(7(y-2))/(2lamda)=(z-3)/(2)and(-7(x-1))/(3lamda)=(y-5)/(1)=(-(z-6))/(5)` are perpendicular to each other.

Options

A. `(70)/(11)`
B. `(7)/(11)`
C. `(17)/(11)`
D. `(71)/(11)`

Answer

To find the value of \( \lambda \) such that the given lines are perpendicular to each other, we will follow these steps:

### Step 1: Write the equations of the lines in parametric form
The equations of the lines are given as:
1. \(-\frac{x-1}{3} = \frac{7(y-2)}{2\lambda} = \frac{z-3}{2}\)
2. \(-\frac{7(x-1)}{3\lambda} = \frac{y-5}{1} = -\frac{(z-6)}{5}\)

We can express these in parametric form.

For the first line, let \( t \) be the parameter:
\[ x = 1 - 3t, \quad y = 2 + \frac{2\lambda}{7}t, \quad z = 3 + 2t \]

For the second line, let \( s \) be the parameter:
\[ x = 1 - \frac{3\lambda}{7}s, \quad y = 5 + s, \quad z = 6 - 5s \]

### Step 2: Identify the direction ratios of both lines
From the parametric equations, we can identify the direction ratios:

For the first line:
- Direction ratios \( a_1 = -3, b_1 = \frac{2\lambda}{7}, c_1 = 2 \)

For the second line:
- Direction ratios \( a_2 = -\frac{3\lambda}{7}, b_2 = 1, c_2 = -5 \)

### Step 3: Use the condition for perpendicularity
Two lines are perpendicular if the dot product of their direction ratios is zero. Thus, we need to set up the equation:
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = 0 \]

Substituting the values:
\[ (-3) \left(-\frac{3\lambda}{7}\right) + \left(\frac{2\lambda}{7}\right)(1) + (2)(-5) = 0 \]

### Step 4: Simplify the equation
Expanding the equation gives:
\[ \frac{9\lambda}{7} + \frac{2\lambda}{7} - 10 = 0 \]
Combining the terms:
\[ \frac{11\lambda}{7} - 10 = 0 \]

### Step 5: Solve for \( \lambda \)
To isolate \( \lambda \):
\[ \frac{11\lambda}{7} = 10 \]
Multiplying both sides by 7:
\[ 11\lambda = 70 \]
Dividing by 11:
\[ \lambda = \frac{70}{11} \]

### Final Answer
Thus, the value of \( \lambda \) is:
\[ \lambda = \frac{70}{11} \]