How many arbitary constants are there in the equation of a plane ?

Options

A. 2
B. 3
C. 4
D. Any finite number

Answer

To determine how many arbitrary constants are present in the equation of a plane, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Standard Equation of a Plane**:
The standard equation of a plane in three-dimensional space can be expressed in the form:
\[ ax + by + cz + d = 0 \]
where \(a\), \(b\), \(c\), and \(d\) are constants.

2. **Identify the Constants**:
In the equation \(ax + by + cz + d = 0\), we can identify four constants: \(a\), \(b\), \(c\), and \(d\).

3. **Understanding Arbitrary Constants**:
An arbitrary constant is a constant whose value can be chosen freely without affecting the overall structure of the equation. In the context of the plane equation, we can manipulate the constants to express the plane in different forms.

4. **Simplifying the Equation**:
We can rearrange the equation to isolate \(d\):
\[ ax + by + cz = -d \]
This shows that \(d\) can be expressed in terms of \(a\), \(b\), and \(c\).

5. **Dividing by a Constant**:
If we assume \(d \neq 0\), we can divide the entire equation by \(-d\):
\[ \frac{a}{-d}x + \frac{b}{-d}y + \frac{c}{-d}z = 1 \]
Let’s denote:
\[ k_1 = \frac{a}{-d}, \quad k_2 = \frac{b}{-d}, \quad k_3 = \frac{c}{-d} \]
This gives us a new representation of the plane:
\[ k_1x + k_2y + k_3z = 1 \]

6. **Counting the Arbitrary Constants**:
From the new representation, we see that we have three arbitrary constants: \(k_1\), \(k_2\), and \(k_3\).

7. **Conclusion**:
Therefore, the number of arbitrary constants in the equation of a plane is **3**.

### Final Answer:
The number of arbitrary constants in the equation of a plane is **3**.

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