A rod of length 2 units whose one ends is `(1, 0, -1)` and other end touches the plane `x-2y+ 2z+4=0`, then which statement is false

Options

A. the rod sweeps the figure whose volume is `pi` cubic units.
B. the area of the region which the rod traces on the plane is `2pi`.
C. the length of projection of the rod on the plane is `sqrt3` units.
D. the centre of the region which the rod traces on the plane is `((2)/(3), (2)/(3), (-5)/(3))`

Explanation

To solve the problem step by step, we need to analyze the given information and find out which statement is false regarding the rod and the plane.

### Step 1: Understand the Problem
We have a rod of length 2 units with one end at the point \( A(1, 0, -1) \) and the other end touching the plane defined by the equation \( x - 2y + 2z + 4 = 0 \).

### Step 2: Find the Distance from Point A to the Plane
To find the distance \( d \) from the point \( A(1, 0, -1) \) to the plane \( x - 2y + 2z + 4 = 0 \), we use the distance formula from a point to a plane:

\[ d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

Where \( A = 1, B = -2, C = 2, D = 4 \) and \( (x_1, y_1, z_1) = (1, 0, -1) \).

Substituting the values:

\[ d = \frac{|1(1) + (-2)(0) + 2(-1) + 4|}{\sqrt{1^2 + (-2)^2 + 2^2}} \]
\[ = \frac{|1 + 0 - 2 + 4|}{\sqrt{1 + 4 + 4}} = \frac{|3|}{\sqrt{9}} = \frac{3}{3} = 1 \]

### Step 3: Determine the Position of the Other End of the Rod
Since the rod is 2 units long and one end is at point \( A(1, 0, -1) \), the other end \( B(x, y, z) \) must be 1 unit away from point \( A \) in the direction perpendicular to the plane.

### Step 4: Find the Radius of the Cylinder Swept by the Rod
The radius \( r \) of the cylinder formed by the rod can be found using the Pythagorean theorem. The length of the rod is the hypotenuse, and the distance from point \( A \) to the plane is one leg of the triangle.

Let \( h \) be the height (distance to the plane) and \( r \) be the radius:

\[ \text{Length of the rod}^2 = r^2 + h^2 \]
\[ 2^2 = r^2 + 1^2 \]
\[ 4 = r^2 + 1 \implies r^2 = 3 \implies r = \sqrt{3} \]

### Step 5: Calculate the Volume of the Cylinder
The volume \( V \) of the cylinder swept by the rod is given by:

\[ V = \pi r^2 h \]
Substituting \( r^2 = 3 \) and \( h = 1 \):

\[ V = \pi \cdot 3 \cdot 1 = 3\pi \]

### Step 6: Check Each Statement
1. **Statement A**: The volume is \( \pi \) cubic units. **(False)**
2. **Statement B**: The area of the region traced by the rod on the plane is \( 2\pi \). **(False)**
3. **Statement C**: The length of the projection of the rod on the plane is \( \sqrt{3} \). **(True)**
4. **Statement D**: The center of the region traced by the rod on the plane is at \( (1, 0, -1) \). **(True)**

### Conclusion
The false statements are A and B. However, since the question asks for which statement is false, we can conclude that:

**The false statement is:** The volume of the rod is \( \pi \) cubic units.