Consider a set of point R in which is at a distance of 2 units from the line `(x)/(1)= (y-1)/(-1)= (z+2)/(2)` between the planes `x-y+2z=3=0 and x-y+2z-2=0`. (a) The volume of the bounded figure by points R and the planes is `(10/3sqrt3)pi` cube units (b) The area of the curved surface formed by the set of points R is `(20/sqrt6)pi` sq. units The volume of the bounded figure by the set of points R and the planes is `(20/sqrt6)pi` cubic units. (d) The area of the curved surface formed by the set of points R is `(10/sqrt3)pi` sq. units

Options

A. The volume of the bounded figure by points R and the planes is `(10//3sqrt3)pi` cube units.
B. The area of the curved surface formed by the set of points R is `(20 pi //sqrt6)` sq. units.
C. The volume of the bounded figure by the set of points R and the planes is `(20pi//sqrt6)` cubic units.
D. The area of the curved surface formed by the set of points R is `(10//sqrt3)pi` sq. units.

Explanation

To solve the problem step by step, we need to analyze the given information regarding the line, the planes, and the distance from the line to the points in space.

### Step 1: Identify the equations of the planes and the line
The equations of the planes are given as:
1. \( x - y + 2z + 3 = 0 \)
2. \( x - y + 2z - 2 = 0 \)

The equation of the line is represented in symmetric form as:
\[ \frac{x}{1} = \frac{y - 1}{-1} = \frac{z + 2}{2} \]

### Step 2: Determine the distance between the two parallel planes
To find the distance \(d\) between two parallel planes given by the equations \(Ax + By + Cz + D_1 = 0\) and \(Ax + By + Cz + D_2 = 0\), we use the formula:
\[ d = \frac{|D_2 - D_1|}{\sqrt{A^2 + B^2 + C^2}} \]
For our planes:
- \(D_1 = 3\)
- \(D_2 = -2\)
- Coefficients \(A = 1\), \(B = -1\), \(C = 2\)

Calculating the distance:
\[ d = \frac{|(-2) - 3|}{\sqrt{1^2 + (-1)^2 + 2^2}} = \frac{|-5|}{\sqrt{1 + 1 + 4}} = \frac{5}{\sqrt{6}} \]

### Step 3: Identify the radius of the cylinder
The problem states that the set of points \(R\) is at a distance of 2 units from the line. This distance acts as the radius \(r\) of the cylinder formed by these points:
\[ r = 2 \]

### Step 4: Calculate the height of the cylinder
The height \(h\) of the cylinder is equal to the distance between the two planes, which we calculated as:
\[ h = \frac{5}{\sqrt{6}} \]

### Step 5: Calculate the volume of the cylinder
The volume \(V\) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the values of \(r\) and \(h\):
\[ V = \pi (2^2) \left(\frac{5}{\sqrt{6}}\right) = \pi \cdot 4 \cdot \frac{5}{\sqrt{6}} = \frac{20\pi}{\sqrt{6}} \]

### Step 6: Calculate the curved surface area of the cylinder
The curved surface area \(A\) of a cylinder is given by the formula:
\[ A = 2\pi rh \]
Substituting the values of \(r\) and \(h\):
\[ A = 2\pi (2) \left(\frac{5}{\sqrt{6}}\right) = 4\pi \cdot \frac{5}{\sqrt{6}} = \frac{20\pi}{\sqrt{6}} \]

### Summary of Results
- Volume of the bounded figure by points \(R\) and the planes: \(\frac{20\pi}{\sqrt{6}}\) cubic units.
- Area of the curved surface formed by the set of points \(R\): \(\frac{20\pi}{\sqrt{6}}\) square units.