What is the distance of the line `2x+y+2z=3` from the origin?

Options

A. 1 unit
B. 1.5 unit
C. 2 unit
D. 2.5 unit

Answer

To find the distance of the line represented by the equation \(2x + y + 2z = 3\) from the origin, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Plane Equation**:
The given line can be interpreted as a plane equation. We can rewrite it in the standard form of a plane:
\[ 2x + y + 2z - 3 = 0 \]
Here, \(A = 2\), \(B = 1\), \(C = 2\), and \(D = -3\).

2. **Identify the Point**:
The point from which we want to find the distance is the origin, which has coordinates:
\[ (x_1, y_1, z_1) = (0, 0, 0) \]

3. **Use the Distance Formula**:
The formula for the distance \(D\) from a point \((x_1, y_1, z_1)\) to a plane \(Ax + By + Cz + D = 0\) is given by:
\[ D = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

4. **Substitute the Values**:
Substitute \(A\), \(B\), \(C\), \(D\), and the coordinates of the origin into the formula:
\[ D = \frac{|2(0) + 1(0) + 2(0) - 3|}{\sqrt{2^2 + 1^2 + 2^2}} \]

5. **Calculate the Numerator**:
The numerator simplifies to:
\[ |0 + 0 + 0 - 3| = |-3| = 3 \]

6. **Calculate the Denominator**:
The denominator simplifies to:
\[ \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

7. **Final Calculation**:
Now, substitute the values back into the distance formula:
\[ D = \frac{3}{3} = 1 \]

### Conclusion:
The distance of the line \(2x + y + 2z = 3\) from the origin is \(1\).

---