Directrix of a parabola is x + y = 2. If it’s focus is origin, then latus rectum of the parabola is equal to

Options

A. `sqrt2` units
B. 2 units
C. `2sqrt2` units
D. 4 units

Answer

To solve the problem, we need to find the length of the latus rectum of a parabola given its directrix and focus. The directrix is given as \( x + y = 2 \) and the focus is at the origin \( (0, 0) \).

### Step-by-Step Solution:

1. **Identify the Directrix and Focus**:
- The directrix is given by the equation \( x + y = 2 \).
- The focus is at the point \( (0, 0) \).

2. **Convert the Directrix Equation**:
- We can rewrite the directrix equation in the form \( Ax + By + C = 0 \):
\[ x + y - 2 = 0 \]
- Here, \( A = 1, B = 1, C = -2 \).

3. **Calculate the Perpendicular Distance from the Focus to the Directrix**:
- The formula for the perpendicular distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
- Substituting \( (x_0, y_0) = (0, 0) \):
\[ d = \frac{|1 \cdot 0 + 1 \cdot 0 - 2|}{\sqrt{1^2 + 1^2}} = \frac{|-2|}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

4. **Relate the Perpendicular Distance to the Latus Rectum**:
- The distance \( d \) from the focus to the directrix is equal to \( 2a \), where \( a \) is the distance from the vertex to the focus.
- Therefore, we have:
\[ 2a = \sqrt{2} \implies a = \frac{\sqrt{2}}{2} \]

5. **Calculate the Length of the Latus Rectum**:
- The length of the latus rectum \( L \) of a parabola is given by the formula:
\[ L = 4a \]
- Substituting the value of \( a \):
\[ L = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \]

### Final Answer:
The length of the latus rectum of the parabola is \( 2\sqrt{2} \) units.

---