The equation `by+cz+d=0` represents a plane parallel to which one of the following ?

Options

A. x- axis
B. y - axis
C. z - axis
D. None of these

Answer

To determine which axis the plane represented by the equation \(by + cz + d = 0\) is parallel to, we can analyze the equation step by step.

### Step-by-Step Solution:

1. **Understand the Equation of the Plane**:
The equation \(by + cz + d = 0\) can be rearranged to express \(z\) in terms of \(y\):
\[ cz = -by - d \implies z = -\frac{b}{c}y - \frac{d}{c} \]
This shows that \(z\) is dependent on \(y\) but does not depend on \(x\).

2. **Identify the Variables**:
In the equation, we see that \(x\) does not appear. This indicates that the value of \(x\) can be any real number while \(y\) and \(z\) are related through the equation.

3. **Determine the Plane Orientation**:
Since \(x\) is not present in the equation, the plane can take any value of \(x\) while \(y\) and \(z\) are constrained by the linear relationship. This means that the plane extends infinitely along the \(x\)-direction.

4. **Conclusion about Parallelism**:
A plane that does not depend on \(x\) and is defined by \(y\) and \(z\) is parallel to the \(yz\)-plane. Therefore, the plane represented by the equation \(by + cz + d = 0\) is parallel to the \(yz\)-plane.

5. **Final Answer**:
The correct answer is **none of these** since the options provided do not include the \(yz\)-plane.