If the line `(x-1)/(1)=(y-k)/(-2)=(z-3)/(lambda)` lies in the plane `3x+4y-2z=6`, then `5|k|+3|lambda|` is equal to

Options

A. 75
B. `(75)/(4)`
C. 15
D. `(5)/(2)`

Answer

To solve the problem, we need to find the values of \( k \) and \( \lambda \) such that the line given by

\[ \frac{x-1}{1} = \frac{y-k}{-2} = \frac{z-3}{\lambda} \]

lies in the plane defined by the equation

\[ 3x + 4y - 2z = 6. \]

### Step 1: Identify the point on the line
From the line equation, we can identify a point on the line, which is \( (1, k, 3) \).

### Step 2: Substitute the point into the plane equation
Since the line lies in the plane, the point \( (1, k, 3) \) must satisfy the plane equation:

\[ 3(1) + 4(k) - 2(3) = 6. \]

### Step 3: Simplify the equation
Substituting the values, we get:

\[ 3 + 4k - 6 = 6. \]

This simplifies to:

\[ 4k - 3 = 6. \]

### Step 4: Solve for \( k \)
Rearranging gives:

\[ 4k = 6 + 3, \]
\[ 4k = 9, \]
\[ k = \frac{9}{4}. \]

### Step 5: Determine the direction ratios
The direction ratios of the line are \( (1, -2, \lambda) \). The normal vector of the plane \( 3x + 4y - 2z = 6 \) is \( (3, 4, -2) \).

### Step 6: Use the dot product condition
For the line to lie in the plane, the direction ratios of the line must be perpendicular to the normal vector of the plane. Therefore, we have:

\[ (1, -2, \lambda) \cdot (3, 4, -2) = 0. \]

### Step 7: Calculate the dot product
Calculating the dot product gives:

\[ 1 \cdot 3 + (-2) \cdot 4 + \lambda \cdot (-2) = 0, \]
\[ 3 - 8 - 2\lambda = 0. \]

### Step 8: Solve for \( \lambda \)
Rearranging gives:

\[ -2\lambda = 5, \]
\[ \lambda = -\frac{5}{2}. \]

### Step 9: Calculate \( 5|k| + 3|\lambda| \)
Now we need to calculate:

\[ 5|k| + 3|\lambda| = 5\left|\frac{9}{4}\right| + 3\left|-\frac{5}{2}\right|. \]

Calculating each term:

\[ 5 \cdot \frac{9}{4} = \frac{45}{4}, \]
\[ 3 \cdot \frac{5}{2} = \frac{15}{2} = \frac{30}{4}. \]

### Step 10: Combine the results
Adding these two results gives:

\[ \frac{45}{4} + \frac{30}{4} = \frac{75}{4}. \]

Thus, the final answer is:

\[ \boxed{\frac{75}{4}}. \]