Equation of a Perpendicular Line Through a Given Point

Given a vertical line and a point, we can determine the equation of a line that is perpendicular to the vertical line and passes through the given point. This article will guide you through the process of finding such an equation, detailing the steps and reasoning involved.

Understanding the Given Line

The equation of the given line is -4x^3 0. Simplifying this, we find:

-4x^3 0 implies x frac{3}{4}

This equation represents a vertical line parallel to the y-axis, passing through the point where x frac{3}{4}.

Identifying the Perpendicular Line

A line that is perpendicular to a vertical line is a horizontal line. The general form of a horizontal line is y c, where c is a constant. This horizontal line must pass through the point (8, 5), which is given in the problem.

To find the value of c, we use the coordinates of the given point. Since the point (8, 5) lies on the horizontal line, the y-coordinate of this point must be the constant value of c. Therefore:

c 5 implies y 5

Final Equation of the Perpendicular Line

Thus, the equation of the line that is perpendicular to the vertical line -4x^3 0 and passes through the point (8, 5) is:

y 5

Conclusion

By understanding the properties of perpendicular lines and the given point, we can determine the equation of the line. In this case, the equation of the perpendicular line is simple and straightforward: y 5.

This problem demonstrates the relationship between vertical and horizontal lines and how to find the equation of a line perpendicular to a given line. Understanding these concepts is essential for solving more complex geometry and algebra problems.