The equation of the plane passing through the points ` (3,2,-1) , (3,4,2) and (7,0,6) ` is `5x + 3y -2z=lamda, ` where `lamda` is :

Options

A. 23
B. 21
C. 19
D. 27

Answer

To find the value of \( \lambda \) for the equation of the plane \( 5x + 3y - 2z = \lambda \) that passes through the points \( (3, 2, -1) \), \( (3, 4, 2) \), and \( (7, 0, 6) \), we can follow these steps:

### Step 1: Substitute the first point into the plane equation
We will use the first point \( (3, 2, -1) \) and substitute its coordinates into the equation of the plane.

\[ 5(3) + 3(2) - 2(-1) = \lambda \]

### Step 2: Calculate the left-hand side
Now, we perform the calculations:

\[ 5(3) = 15 \]
\[ 3(2) = 6 \]
\[ -2(-1) = 2 \]

Adding these values together:

\[ 15 + 6 + 2 = \lambda \]

### Step 3: Simplify the expression
Now, we simplify the expression:

\[ 15 + 6 = 21 \]
\[ 21 + 2 = 23 \]

Thus, we find:

\[ \lambda = 23 \]

### Conclusion
The value of \( \lambda \) is \( 23 \).

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