Find the value of `lambda` such that line `(x-2)/(6) = (y-1)/(lambda) = (z+5)/(-4)` is perpendicular to the plane `3x-y-2z=7`.

Answer

To find the value of \( \lambda \) such that the line

\[ \frac{x-2}{6} = \frac{y-1}{\lambda} = \frac{z+5}{-4} \]

is perpendicular to the plane

\[ 3x - y - 2z = 7, \]

we will follow these steps:

### Step 1: Identify the direction ratios of the line
The given line can be expressed in terms of its direction ratios. From the equation:

\[ \frac{x-2}{6} = \frac{y-1}{\lambda} = \frac{z+5}{-4}, \]

the direction ratios of the line are \( (6, \lambda, -4) \).

### Step 2: Identify the normal vector of the plane
The equation of the plane is given as:

\[ 3x - y - 2z = 7. \]

The coefficients of \( x, y, z \) in this equation give us the normal vector of the plane, which is \( (3, -1, -2) \).

### Step 3: Use the condition for perpendicularity
For the line to be perpendicular to the plane, the direction ratios of the line must be proportional to the normal vector of the plane. This means we can set up the following relationship:

\[ \frac{6}{3} = \frac{\lambda}{-1} = \frac{-4}{-2}. \]

### Step 4: Solve the proportions
From the first part of the proportion:

\[ \frac{6}{3} = 2. \]

From the second part of the proportion:

\[ \frac{-4}{-2} = 2. \]

Now, we set the middle part equal to 2:

\[ \frac{\lambda}{-1} = 2. \]

### Step 5: Solve for \( \lambda \)
To find \( \lambda \), we multiply both sides by \(-1\):

\[ \lambda = -2. \]

### Final Answer
Thus, the value of \( \lambda \) such that the line is perpendicular to the plane is

\[ \lambda = -2. \]

---