The equation of set of lines which are at a constant distance 2 units from the origin is

Options

A. x+y+2=0
B. x+y+4=0
C. `x "cos" alpha + y "sin" alpha = 2`
D. `x "cos" alpha + y "sin" alpha = (1)/(2)`

Explanation

To find the equation of the set of lines that are at a constant distance of 2 units from the origin, we can use the concept of the distance of a line from a point.

### Step-by-Step Solution:

1. **Understanding the Distance from the Origin**:
The distance \( P \) from the origin (0, 0) to a line can be expressed using the formula:
\[ \text{Distance} = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
where \( Ax + By + C = 0 \) is the equation of the line.

2. **Setting Up the Equation**:
For lines at a constant distance \( P = 2 \) from the origin, we can set up the equation:
\[ \frac{|C|}{\sqrt{A^2 + B^2}} = 2 \]
This implies:
\[ |C| = 2\sqrt{A^2 + B^2} \]

3. **General Form of the Line**:
The general form of the line can be expressed as:
\[ Ax + By + C = 0 \]
We can express \( C \) in terms of \( A \) and \( B \):
\[ C = \pm 2\sqrt{A^2 + B^2} \]

4. **Finding the Set of Lines**:
Substituting \( C \) back into the line equation gives us:
\[ Ax + By \pm 2\sqrt{A^2 + B^2} = 0 \]
This represents two families of lines for each pair of \( A \) and \( B \).

5. **Final Equation**:
Therefore, the equation of the set of lines that are at a constant distance of 2 units from the origin can be represented as:
\[ Ax + By = \pm 2\sqrt{A^2 + B^2} \]
where \( A \) and \( B \) can take any non-zero values.