Find the equation of the straight line which is at a distance 5 units from the origin and the
perpendicular from the origin to the line makes an angle 45◦ with the positive direction of
the x-axis

We need to find the equation of a straight line which is at a distance of 5 units from the origin, and the perpendicular from the origin to the line makes an angle of 45° with the positive direction of the x-axis.

### Step 1: General equation of a straight line in normal form
The equation of a straight line in normal form is given by:

\[
x \cos \theta + y \sin \theta = p
\]

Where:
- \( \theta \) is the angle made by the perpendicular from the origin to the line with the positive x-axis.
- \( p \) is the perpendicular distance from the origin to the line.

### Step 2: Substituting the values
From the problem, we know:
- The distance from the origin \( p = 5 \) units.
- The angle \( \theta = 45^\circ \).

So, substituting \( \theta = 45^\circ \) and \( p = 5 \) into the equation:

\[
x \cos 45^\circ + y \sin 45^\circ = 5
\]

Since \( \cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}} \), the equation becomes:

\[
x \frac{1}{\sqrt{2}} + y \frac{1}{\sqrt{2}} = 5
\]

Multiplying through by \( \sqrt{2} \) to eliminate the fractions:

\[
x + y = 5\sqrt{2}
\]

### Final equation
Thus, the equation of the straight line is:

\[
x + y = 5\sqrt{2}
\]

### Conclusion
The required equation of the line is \( x + y = 5\sqrt{2} \).

Would you like further details or have any questions? Here are 5 questions to extend your understanding:

1. How do we derive the normal form of a line equation?
2. Why do \( \cos 45^\circ \) and \( \sin 45^\circ \) have the same value?
3. What happens to the equation of the line if the distance from the origin is changed?
4. How does the angle of the perpendicular influence the orientation of the line?
5. What are other forms of line equations, and when are they useful?

**Tip**: When working with line equations, always confirm which form (slope-intercept, normal, etc.) suits your problem best for easier computation.

Find the equation of the straight line which is at a distance 5 units from the origin and the perpendicular from the origin to the line makes an angle 45◦ with the positive direction of the x-axis.

Discover how to find the equation of a straight line that is at a distance of 5 units from the origin, with the perpendicular making a 45° angle with the positive x-axis. This problem covers key concepts in geometry and coordinate geometry, perfect for students in Grades 9-11.

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