The two planes `ax+by+cz+d=0 and ax+by+cz+d_1=0` where `dned_1` have

Options

A. A) One point only in common
B. B) Three points in common
C. C) Infinite points in common
D. D) No point in common

Answer

To solve the problem of finding the number of common points between the two planes given by the equations \( ax + by + cz + d = 0 \) and \( ax + by + cz + d_1 = 0 \) where \( d \neq d_1 \), we can follow these steps:

### Step 1: Write down the equations of the planes
We have two planes represented by the equations:
1. \( ax + by + cz + d = 0 \) (Equation 1)
2. \( ax + by + cz + d_1 = 0 \) (Equation 2)

### Step 2: Identify the coefficients
From the equations, we can identify the coefficients:
- For Equation 1: \( A_1 = a, B_1 = b, C_1 = c \)
- For Equation 2: \( A_2 = a, B_2 = b, C_2 = c \)

### Step 3: Check the ratios of the coefficients
To determine if the planes are parallel, we check the ratios of the coefficients:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]
Substituting the values:
\[ \frac{a}{a} = \frac{b}{b} = \frac{c}{c} \]
This simplifies to:
\[ 1 = 1 = 1 \]
This means that the ratios are equal.

### Step 4: Conclusion about the planes
Since the ratios of the coefficients are equal, we conclude that the two planes are parallel.

### Step 5: Determine the number of common points
For two parallel planes, there are no points of intersection. Therefore, the number of common points between the two planes is zero.

### Final Answer
The number of common points between the two planes is **0**.

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