Understanding Perpendicular and Parallel Lines in Coordinate Geometry

In coordinate geometry, understanding the properties of perpendicular and parallel lines is fundamental. This article will guide you through the concepts of finding the equations of lines that are either perpendicular or parallel to a given line, using specific examples and step-by-step methods.

Perpendicular Lines

What is the equation of a line perpendicular to y -3x - 2 passing through the point (9, -2)?

Step 1: Determine the slope of the original line. The slope of the line y -3x - 2 is -3.

Step 2: Find the slope of the perpendicular line. The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line is m 1/3.

Step 3: Use the point-slope formula to find the equation of the perpendicular line.

y - y1 m(x - x1), where (x1, y1) (9, -2) and m 1/3.

y - (-2) 1/3(x - 9) y 2 1/3(x - 9)

Multiplying both sides by 3 to clear the fraction: 3y 6 x - 9 3y x - 15 y 1/3x - 5

Therefore, the equation of the line perpendicular to y -3x - 2 and passing through the point (9, -2) is y 1/3x - 5.

Parallel Lines

Now, let's explore the concept of parallel lines. Given a line y -3x - 2, we want to find the equation of a line parallel to this and passing through the point (9, -2).

Step 1: Determine the slope of the given line. The slope of the line y -3x - 2 is -3.

Step 2: Use the same slope for the parallel line since parallel lines have the same slope. Therefore, the slope of the parallel line is still -3.

Step 3: Use the point-slope formula to find the equation of the parallel line.

y - y1 m(x - x1), where (x1, y1) (9, -2) and m -3.

y - (-2) -3(x - 9) y 2 -3(x - 9)

Multiplying both sides by -3 to clear the fraction: y 2 -3x 27 y -3x 27 - 2 y -3x 25

Therefore, the equation of the line parallel to y -3x - 2 and passing through the point (9, -2) is y -3x 25.

Conclusion

Understanding the differences and relationships between perpendicular and parallel lines in coordinate geometry is crucial. By following these steps and formulas, you can easily determine the equations of both types of lines. Whether you're working with perpendicular or parallel lines, the key is to remember the slope and how it affects the line equation.