To find the equation of the plane that passes through the points (2, 2, 0) and (4, 4, 0) and is parallel to the Z-axis, we can follow these steps:
### Step 1: Understand the properties of the plane
Since the plane is parallel to the Z-axis, it means that the Z-coordinate does not affect the equation of the plane. Therefore, the equation will only involve the X and Y coordinates.
**Hint:** Remember that a plane parallel to the Z-axis does not change with respect to Z.
### Step 2: Use the general equation of a plane
The general equation of a plane can be expressed as:
\[ A(x - x_0) + B(y - y_0) + C(z - z_0) = 0 \]
where \((x_0, y_0, z_0)\) is a point on the plane. Since we have two points, we can use one of them, say (2, 2, 0).
**Hint:** Choose one of the given points to simplify the equation.
### Step 3: Substitute the point into the plane equation
Using the point (2, 2, 0), we can substitute into the equation:
\[ A(x - 2) + B(y - 2) + C(z - 0) = 0 \]
Since the plane is parallel to the Z-axis, we can set \(C = 0\). Thus, the equation simplifies to:
\[ A(x - 2) + B(y - 2) = 0 \]
**Hint:** Setting \(C = 0\) is crucial for planes parallel to the Z-axis.
### Step 4: Use the second point to find the relationship between A and B
Now, substitute the second point (4, 4, 0) into the equation:
\[ A(4 - 2) + B(4 - 2) = 0 \]
This simplifies to:
\[ 2A + 2B = 0 \]
Dividing through by 2 gives:
\[ A + B = 0 \]
Thus, we can express \(B\) in terms of \(A\):
\[ B = -A \]
**Hint:** This relationship helps us express one variable in terms of the other.
### Step 5: Substitute back to find the equation of the plane
Now substitute \(B = -A\) back into the equation:
\[ A(x - 2) - A(y - 2) = 0 \]
Factoring out \(A\) gives:
\[ A[(x - 2) - (y - 2)] = 0 \]
This simplifies to:
\[ A(x - y) = 0 \]
Since \(A\) cannot be zero (otherwise we wouldn't have a plane), we can conclude:
\[ x - y = 0 \]
**Hint:** The equation \(x - y = 0\) indicates that the plane is vertical and extends infinitely along the Z-axis.
### Final Answer:
The equation of the plane passing through the points (2, 2, 0) and (4, 4, 0) and parallel to the Z-axis is:
\[ x - y = 0 \]
