Consider the three planes
`P_1 :3x +15 y +21 z=9`,
` P_2 :x - 3y -z=5 ,` and
` P_3 : 2x +10y +14 z =5`
then , which one of the following is true ?

Options

A. `P_1 and P_2` are parallel
B. `P_1 and P_3 ` are parallel
C. `P_2 and P_3` are parallel
D. `P_1 , P_2` and ` P_3` all are parallel

Answer

To determine the relationship between the three given planes \( P_1, P_2, \) and \( P_3 \), we will analyze their equations step by step.

### Step 1: Write down the equations of the planes
The equations of the planes are:
1. \( P_1: 3x + 15y + 21z = 9 \)
2. \( P_2: x - 3y - z = 5 \)
3. \( P_3: 2x + 10y + 14z = 5 \)

### Step 2: Simplify the equations
We can simplify the equations to make it easier to compare them.

- For \( P_1 \):
\[ 3x + 15y + 21z = 9 \implies x + 5y + 7z = 3 \quad \text{(dividing the entire equation by 3)} \]

- For \( P_2 \):
\[ x - 3y - z = 5 \quad \text{(already simplified)} \]

- For \( P_3 \):
\[ 2x + 10y + 14z = 5 \implies x + 5y + 7z = \frac{5}{2} \quad \text{(dividing the entire equation by 2)} \]

Now we have:
1. \( P_1: x + 5y + 7z = 3 \)
2. \( P_2: x - 3y - z = 5 \)
3. \( P_3: x + 5y + 7z = \frac{5}{2} \)

### Step 3: Compare the coefficients for parallelism
To check if two planes are parallel, we need to compare the coefficients of \( x, y, z \) in their equations.

- For \( P_1 \) and \( P_3 \):
- Coefficients of \( P_1 \): \( (1, 5, 7) \)
- Coefficients of \( P_3 \): \( (1, 5, 7) \)

The ratios of the coefficients are:
\[ \frac{1}{1} = \frac{5}{5} = \frac{7}{7} = 1 \]
Since the ratios are equal, \( P_1 \) and \( P_3 \) are parallel.

### Step 4: Check if \( P_2 \) is parallel to any other plane
Now we check \( P_2 \):
- Coefficients of \( P_2 \): \( (1, -3, -1) \)

Comparing \( P_2 \) with \( P_1 \) and \( P_3 \):
- For \( P_1 \) and \( P_2 \):
- Ratios: \( \frac{1}{1}, \frac{5}{-3}, \frac{7}{-1} \) are not equal.

- For \( P_2 \) and \( P_3 \):
- Ratios: \( \frac{1}{1}, \frac{-3}{5}, \frac{-1}{7} \) are not equal.

Thus, \( P_2 \) is not parallel to either \( P_1 \) or \( P_3 \).

### Conclusion
The only conclusion we can draw is that \( P_1 \) and \( P_3 \) are parallel planes.

### Final Answer
The correct option is that \( P_1 \) and \( P_3 \) are parallel.

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