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To solve this problem, we need to find the equation of a line based on two pieces of information:
1. The perpendicular distance from the origin to the line is \( 2 \) units.
2. The perpendicular makes an angle of \( 45^\circ \) with the positive direction of the x-axis.

### Solution:
The general form of the equation of a line based on its normal form is:
\[
x \cos \theta + y \sin \theta = p
\]
where:
- \( \theta \) is the angle the perpendicular makes with the positive x-axis.
- \( p \) is the length of the perpendicular from the origin to the line.

For this problem:
- \( \theta = 45^\circ \)
- \( p = 2 \)

1. Substitute \( \theta = 45^\circ \) and \( p = 2 \) into the equation:
   \[
   x \cos 45^\circ + y \sin 45^\circ = 2
   \]

2. We know that \( \cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2} \), so:
   \[
   x \cdot \frac{\sqrt{2}}{2} + y \cdot \frac{\sqrt{2}}{2} = 2
   \]

3. Simplify the equation:
   \[
   \frac{\sqrt{2}}{2} (x + y) = 2
   \]

4. Multiply both sides by \( \frac{2}{\sqrt{2}} \) to eliminate the fraction:
   \[
   x + y = 2\sqrt{2}
   \]

Therefore, the equation of the line is:
\[
x + y = 2\sqrt{2}
\]

Would you like a further breakdown of any step? 

Here are some related questions you might explore:
1. What is the general form of a line equation when given the slope and a point?
2. How would the equation change if the perpendicular length were different?
3. What happens to the line equation if the angle with the x-axis is \( 90^\circ \)?
4. How can you find the distance between a point and a line?
5. What is the geometric interpretation of the normal form of a line?

**Tip:** Knowing how to convert between different forms of a line equation is a valuable skill in geometry and coordinate algebra.

Determine the equation of the line whose perpendicular from the origin is of length 2 units and makes an angle of 45° with the positive direction of x-axis.

Solve the equation of a line whose perpendicular distance from the origin is 2 units, making an angle of 45° with the x-axis. This detailed solution covers the normal form of a line equation and trigonometric values for 45° angles. Ideal for students in Grades 9-11.

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