A plane P passes through the line of intersection of the planes `2x-y+3z=2,x+y-z=1` and the point (1,0,1).
What is the equation of the plane P?

Options

A. `2x+5y-2=0`
B. `5x+2y-5=0`
C. `x+z-2=0`
D. `2x-y-2z=0`

Explanation

To find the equation of the plane \( P \) that passes through the line of intersection of the planes \( 2x - y + 3z = 2 \) and \( x + y - z = 1 \), and also through the point \( (1, 0, 1) \), we can follow these steps:

### Step 1: Write down the equations of the given planes.
The equations of the planes are:
1. Plane 1: \( 2x - y + 3z - 2 = 0 \) (let's call this \( P_1 \))
2. Plane 2: \( x + y - z - 1 = 0 \) (let's call this \( P_2 \))

### Step 2: Use the equation of the plane through the intersection of two planes.
The general equation of a plane passing through the line of intersection of two planes \( P_1 \) and \( P_2 \) can be expressed as:
\[ P: P_1 + \lambda P_2 = 0 \]
Substituting the equations of \( P_1 \) and \( P_2 \):
\[ (2x - y + 3z - 2) + \lambda (x + y - z - 1) = 0 \]

### Step 3: Expand the equation.
Expanding this gives:
\[ 2x - y + 3z - 2 + \lambda x + \lambda y - \lambda z - \lambda = 0 \]
Combining like terms results in:
\[ (2 + \lambda)x + (-1 + \lambda)y + (3 - \lambda)z - (2 + \lambda) = 0 \]

### Step 4: Substitute the point (1, 0, 1) into the equation.
Since the plane passes through the point \( (1, 0, 1) \), we substitute \( x = 1 \), \( y = 0 \), and \( z = 1 \) into the equation:
\[ (2 + \lambda)(1) + (-1 + \lambda)(0) + (3 - \lambda)(1) - (2 + \lambda) = 0 \]
This simplifies to:
\[ (2 + \lambda) + (3 - \lambda) - (2 + \lambda) = 0 \]
Combining terms gives:
\[ 3 = 0 \]
This means we need to simplify further:
\[ 2 + \lambda + 3 - \lambda - 2 - \lambda = 0 \]
This simplifies to:
\[ 3 - \lambda = 0 \]
Thus, we find:
\[ \lambda = 3 \]

### Step 5: Substitute \( \lambda \) back into the plane equation.
Now substitute \( \lambda = 3 \) back into the equation of the plane:
\[ (2 + 3)x + (-1 + 3)y + (3 - 3)z - (2 + 3) = 0 \]
This simplifies to:
\[ 5x + 2y + 0z - 5 = 0 \]
Or:
\[ 5x + 2y - 5 = 0 \]

### Final Equation of the Plane
Thus, the equation of the plane \( P \) is:
\[ 5x + 2y - 5 = 0 \]