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°

The equation of a straight line in normal form is given by:

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

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

### Given:
- The length of the perpendicular \( p = 4 \) units,
- The inclination of the perpendicular \( \theta = 135^\circ \).

Substituting these values into the equation:

\[
x \cos 135^\circ + y \sin 135^\circ = 4
\]

We know:
- \( \cos 135^\circ = -\frac{1}{\sqrt{2}} \),
- \( \sin 135^\circ = \frac{1}{\sqrt{2}} \).

Thus, the equation becomes:

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

Multiplying the entire equation by \( \sqrt{2} \) to eliminate the denominators:

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

Therefore, the equation of the straight line is:

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

Or equivalently:

\[
x - y + 4\sqrt{2} = 0
\]

This is the required equation of the straight line.

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. What is the equation of a line if \( p = 5 \) and \( \theta = 45^\circ \)?
2. How does the equation of a line change if \( \theta = 90^\circ \)?
3. Find the equation of the line with \( p = 3 \) and \( \theta = 30^\circ \).
4. What happens to the line if the perpendicular length increases but \( \theta \) remains the same?
5. How would the equation change if the angle \( \theta \) were negative?

**Tip:** The normal form of the equation of a line is a useful way to define the line geometrically by focusing on the perpendicular distance and angle, rather than the usual slope-intercept form.

Find the equation of a line where the perpendicular from the origin is 4 units, and the angle to the X-axis is 135°. This solution explains the normal form of a line equation and includes step-by-step instructions using trigonometric values for cos(135°) and sin(135°). Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation