Given planes `P_1:cy+bz=x`
`P_2:az+cx=y`
`P_3:bx+ay=z`
`P_1, P_2 and P_3` pass through one line, if

Options

A. `a^2+b^2+c^2=ab+bc+ca`
B. `a^2+b^2+c^2+2abc=1`
C. `a^2+b^2+c^2=1`
D. `a^2+b^2+c^2+2ab+2bc+2ca+2abc=1`

Answer

To solve the problem, we need to determine the condition under which the three given planes \( P_1: cy + bz = x \), \( P_2: az + cx = y \), and \( P_3: bx + ay = z \) intersect along a line.

### Step-by-Step Solution:

1. **Write the equations of the planes**:
- Plane \( P_1: cy + bz = x \) can be rearranged to \( x - cy - bz = 0 \).
- Plane \( P_2: az + cx = y \) can be rearranged to \( y - az - cx = 0 \).
- Plane \( P_3: bx + ay = z \) can be rearranged to \( z - bx - ay = 0 \).

2. **Multiply the equations**:
- Multiply the equation of \( P_1 \) by \( A \):
\[ A(x - cy - bz) = 0 \implies Ax - Acy - Abz = 0 \]
- Multiply the equation of \( P_3 \) by \( B \):
\[ B(z - bx - ay) = 0 \implies Bz - Bbx - Bay = 0 \]

3. **Add the modified equations**:
- Adding the modified equations from step 2:
\[ Ax - Acy - Abz + Bz - Bbx - Bay = 0 \]
- Rearranging gives:
\[ (A - Bb)x + (Acy + Bay) + (B - Ab)z = 0 \]

4. **Set up the conditions for a line of intersection**:
- For the planes to intersect along a line, the coefficients of \( x \), \( y \), and \( z \) must be dependent. This leads to the following relationships:
\[ A - Bb = 0, \quad Acy + Bay + (B - Ab)z = 0 \]

5. **Use the second plane**:
- Similarly, multiply \( P_2 \) by \( A \) and \( P_3 \) by \( B \) and perform similar operations to derive another equation.

6. **Equate the ratios**:
- From the equations derived, we can express \( x/y \) and \( y/z \) in terms of \( A \), \( B \), and \( C \):
\[ \frac{x}{y} = \frac{ab + c}{1 - b^2}, \quad \frac{y}{z} = \frac{1 - a^2}{ab + c} \]

7. **Cross-multiply and simplify**:
- Set the two ratios equal to each other:
\[ \frac{ab + c}{1 - b^2} = \frac{1 - a^2}{ab + c} \]
- Cross-multiplying leads to:
\[ (ab + c)^2 = (1 - a^2)(1 - b^2) \]

8. **Final condition**:
- Expanding both sides results in:
\[ a^2 + b^2 + c^2 + 2abc = 1 \]
- This is the condition for the planes to intersect along a line.

### Conclusion:
The condition for the planes \( P_1, P_2, \) and \( P_3 \) to intersect along a line is given by:
\[ a^2 + b^2 + c^2 + 2abc = 1 \]