Find the equation of the straight line such that the length of the perpendicular from origin is 4 units and the inclination of this perpendicular to X-axis is 135°

To find the equation of the straight line, given that:

- The length of the perpendicular from the origin is 4 units.
- The inclination of this perpendicular to the X-axis is 135°.

We can use the **normal form** of the equation of a straight line. The general form of this is:

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

Where:
- \( \theta \) is the angle of inclination of the perpendicular from the origin to the line (with respect to the positive X-axis),
- \( p \) is the length of this perpendicular.

### Step 1: Substituting the known values:
- \( p = 4 \) (length of the perpendicular),
- \( \theta = 135^\circ \).

Now, we can calculate the trigonometric values for \( \theta = 135^\circ \):
\[
\cos 135^\circ = -\frac{1}{\sqrt{2}}, \quad \sin 135^\circ = \frac{1}{\sqrt{2}}
\]

### Step 2: Plug these values into the normal form equation:

\[
x \left( -\frac{1}{\sqrt{2}} \right) + y \left( \frac{1}{\sqrt{2}} \right) = 4
\]

### Step 3: Simplifying the equation:
Multiply the whole equation by \( \sqrt{2} \) to eliminate the square root in the denominator:

\[
-x + y = 4\sqrt{2}
\]

### Final Equation:
The equation of the required line is:

\[
x - y = -4\sqrt{2}
\]

Would you like more details or explanations on any of the steps?

Here are 5 questions to expand your understanding:
1. What is the normal form of a straight line in general?
2. How do you convert between different forms of a straight line's equation?
3. What is the significance of the inclination angle of the perpendicular in the normal form?
4. Can you find the equation if the inclination angle were different, such as 45°?
5. How do trigonometric values change for angles like \(135^\circ\)?

**Tip:** When working with trigonometric angles, remember that the values for angles like \(135^\circ\) come from their position in the unit circle in the second quadrant, where cosine is negative and sine is positive.

Find the equation of the straight line such that the length of the perpendicular from origin is 4 units and the inclination of this perpendicular to X-axis is 135°.

Learn how to find the equation of a straight line where the perpendicular from the origin has a length of 4 units and makes an angle of 135° with the X-axis. This detailed solution explains the normal form of the line equation and key trigonometric concepts, suitable for students in Grades 10-12.

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