Which of the following planes are parallel but not identical?
`P_1: 4x-2y+6z=3`
`P_2: 4x-2y-2z=6`
`P_3: -6x+3y-9z=5`
`P_4: 2x-y-z=3`

Options

A. (a)`P_2 and P_3`
B. (b)`P_2 and P_4`
C. (c)`P_1 and P_3`
D. (d)`P_1 and P_4`

Answer

To determine which planes among the given options are parallel but not identical, we need to analyze the equations of the planes. The general form of a plane is given by the equation \(Ax + By + Cz = D\). For two planes to be parallel, the ratios of their coefficients must be equal, but the ratio of the constants must not be equal.

### Given Planes:
1. \(P_1: 4x - 2y + 6z = 3\)
2. \(P_2: 4x - 2y - 2z = 6\)
3. \(P_3: -6x + 3y - 9z = 5\)
4. \(P_4: 2x - y - z = 3\)

### Step 1: Identify Coefficients
For each plane, identify the coefficients \(A\), \(B\), \(C\), and the constant \(D\):
- For \(P_1\): \(A_1 = 4\), \(B_1 = -2\), \(C_1 = 6\), \(D_1 = 3\)
- For \(P_2\): \(A_2 = 4\), \(B_2 = -2\), \(C_2 = -2\), \(D_2 = 6\)
- For \(P_3\): \(A_3 = -6\), \(B_3 = 3\), \(C_3 = -9\), \(D_3 = 5\)
- For \(P_4\): \(A_4 = 2\), \(B_4 = -1\), \(C_4 = -1\), \(D_4 = 3\)

### Step 2: Check Parallelism
We will check pairs of planes to see if they are parallel using the condition:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \quad \text{and} \quad \frac{D_1}{D_2} \text{ should not be equal.} \]

#### Option A: \(P_2\) and \(P_3\)
- Coefficients: \(P_2: (4, -2, -2)\) and \(P_3: (-6, 3, -9)\)
- Ratios:
\[ \frac{4}{-6} \neq \frac{-2}{3} \neq \frac{-2}{-9} \]
- Not parallel.

#### Option B: \(P_2\) and \(P_4\)
- Coefficients: \(P_2: (4, -2, -2)\) and \(P_4: (2, -1, -1)\)
- Ratios:
\[ \frac{4}{2} = 2, \quad \frac{-2}{-1} = 2, \quad \frac{-2}{-1} = 2 \]
- Check constants:
\[ \frac{6}{3} = 2 \quad \text{(identical)} \]
- Not parallel.

#### Option C: \(P_1\) and \(P_3\)
- Coefficients: \(P_1: (4, -2, 6)\) and \(P_3: (-6, 3, -9)\)
- Ratios:
\[ \frac{4}{-6} = -\frac{2}{3}, \quad \frac{-2}{3} = \frac{6}{-9} \]
- Check constants:
\[ \frac{3}{5} \quad \text{(not equal)} \]
- Parallel but not identical.

#### Option D: \(P_1\) and \(P_4\)
- Coefficients: \(P_1: (4, -2, 6)\) and \(P_4: (2, -1, -1)\)
- Ratios:
\[ \frac{4}{2} = 2, \quad \frac{-2}{-1} = 2, \quad \frac{6}{-1} \neq 3 \]
- Not parallel.

### Conclusion
The only pair of planes that are parallel but not identical is:
- **Option C: \(P_1\) and \(P_3\)**.