Introduction

Understanding the concept of a normal line is crucial in various mathematical and engineering applications. A normal line to a given line is a line that intersects the first line at a right angle. This article delves into the process of finding the equation of a line that is normal to a given line, using a specific example as a walkthrough. We will explore the methods for determining the slope of the normal line and applying the point-slope form to find the desired equation.

Understanding the Concept of a Normal Line

A line is considered normal to another line if it intersects the first line at a 90-degree angle. The process of finding the equation of the normal line involves understanding the slopes of both lines and using a point provided to construct the equation.

Given Line and Its Properties

The equation of the given line is x - y 3 0. This can be rewritten in the slope-intercept form as:

y x 3

The slope (m) of this line is 1.

Step-by-Step Solution

1. **Determine the Slope of the Normal Line:**

Since the normal line is perpendicular to the given line, the slope of the normal line (mnormal) can be calculated as the negative reciprocal of the slope of the given line.

Given that the slope (m) of the given line is 1, the slope of the normal line is:

mnormal -1

2. **Using the Point-Slope Form:** [ y - y_1 m(x - x_1) ] **Given Point:** (M3 - 1) [ x_1 3, quad y_1 -1 ] Substituting the values into the point-slope form: [ y - (-1) -1(x - 3) ] [ y 1 -1x 3 ] [ y -x 2 ] [ y frac{1}{2}x - frac{1}{2} ]3. **Alternative Methods:** - **Step 1:** Determine the slope of the given line (x - y 3 0): [ y x 3 ] Slope (m) 1 - **Step 2:** The slope of the normal line is: [ m_{normal} -1 ] - **Step 3:** Using the point-slope form: [ y - (-1) -1(x - 3) ] [ y 1 -x 3 ] [ y -x 2 ]4. **Conclusion:** The equation of the line which passes through the point (M(3, -1)) and is normal to the line (x - y 3 0) is: [boldsymbol{y -x 2}] Alternatively, the equation can be written as: [boldsymbol{y frac{1}{2}x - frac{1}{2}}]

Additional Examples and Practice

For further practice, consider finding the equation of a normal line for other given points and lines. The process remains consistent, requiring the slope of the original line and the use of the point-slope form with the appropriate point.

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