Which Equation Represents The Line That Is Perpendicular To $y=\frac{1}{4}$ And Passes Through $(-6,-9$\]?A. $x=-9$ B. $x=-6$ C. $y=-9$ D. $y=-4$

Introduction

In mathematics, the concept of perpendicular lines is a fundamental aspect of geometry and algebra. When two lines are perpendicular, they intersect at a right angle, forming an "L" shape. In this article, we will explore the equation of a line that is perpendicular to a given line and passes through a specific point. We will use the concept of slope to determine the equation of the perpendicular line.

Understanding Slope

The slope of a line is a measure of how steep it is. It is calculated by dividing the vertical change (rise) by the horizontal change (run). The slope of a line can be represented by the equation:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $m$ is the slope, and $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

The Given Line

The given line is represented by the equation:

$$y = \frac{1}{4}$$

This is a horizontal line with a slope of 0. Since the slope of the given line is 0, the slope of the perpendicular line will be undefined.

The Perpendicular Line

A line with an undefined slope is a vertical line. Therefore, the equation of the perpendicular line will be in the form:

$$x = a$$

where $a$ is a constant.

Passing Through a Point

The perpendicular line must pass through the point $(-6, -9)$. To find the equation of the perpendicular line, we can substitute the coordinates of the point into the equation:

$$x = a$$

$$-6 = a$$

Therefore, the equation of the perpendicular line is:

$$x = -6$$

Conclusion

In conclusion, the equation of the line that is perpendicular to $y = \frac{1}{4}$ and passes through $(-6, -9)$ is $x = -6$. This is a vertical line with an undefined slope.

Answer

The correct answer is:

B. $x = -6$

Discussion

This problem requires a basic understanding of slope and perpendicular lines. The concept of slope is essential in determining the equation of a line. The given line is a horizontal line with a slope of 0, and the perpendicular line is a vertical line with an undefined slope. The equation of the perpendicular line is found by substituting the coordinates of the point into the equation.

Additional Examples

1. Find the equation of the line that is perpendicular to $y = 2x + 1$ and passes through $(3, 7)$.
2. Find the equation of the line that is perpendicular to $x = 4$ and passes through $(2, -3)$.

Solutions

1. The slope of the given line is 2. The slope of the perpendicular line will be the negative reciprocal of 2, which is $-\frac{1}{2}$. The equation of the perpendicular line will be in the form:

$$y = -\frac{1}{2}x + b$$

Substituting the coordinates of the point into the equation, we get:

$$7 = -\frac{1}{2}(3) + b$$

Solving for $b$, we get:

$$b = 8$$

Therefore, the equation of the perpendicular line is:

$$y = -\frac{1}{2}x + 8$$

2. The equation of the line that is perpendicular to $x = 4$ will be in the form:

$$y = a$$

where $a$ is a constant. Since the line passes through the point $(2, -3)$, we can substitute the coordinates of the point into the equation:

$$-3 = a$$

Therefore, the equation of the perpendicular line is:

$$y = -3$$

Conclusion

Q&A: Perpendicular Lines and Slope

Q: What is the slope of a line that is perpendicular to a horizontal line? A: The slope of a line that is perpendicular to a horizontal line is undefined. This is because a horizontal line has a slope of 0, and the negative reciprocal of 0 is undefined.

Q: What is the equation of a line that is perpendicular to a horizontal line? A: The equation of a line that is perpendicular to a horizontal line is a vertical line. It can be represented by the equation:

$$x = a$$

where $a$ is a constant.

Q: How do you find the equation of a line that is perpendicular to a given line and passes through a specific point? A: To find the equation of a line that is perpendicular to a given line and passes through a specific point, you need to follow these steps:

1. Find the slope of the given line.
2. Find the negative reciprocal of the slope of the given line. This will be the slope of the perpendicular line.
3. Use the point-slope form of a line to write the equation of the perpendicular line.

Q: What is the point-slope form of a line? A: The point-slope form of a line is:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope of the line.

Q: How do you find the equation of a line that is perpendicular to a vertical line? A: To find the equation of a line that is perpendicular to a vertical line, you need to follow these steps:

1. Find the equation of the vertical line.
2. The equation of the perpendicular line will be a horizontal line. It can be represented by the equation:

$$y = a$$

where $a$ is a constant.

Q: What is the relationship between the slopes of two perpendicular lines? A: The slopes of two perpendicular lines are negative reciprocals of each other. This means that if the slope of one line is $m$, the slope of the perpendicular line will be $-\frac{1}{m}$.

Q: How do you determine if two lines are perpendicular? A: To determine if two lines are perpendicular, you need to check if the product of their slopes is -1. If the product of their slopes is -1, then the lines are perpendicular.

Q: What is the equation of a line that is perpendicular to the line $y = 2x + 1$ and passes through the point $(3, 7)$? A: To find the equation of a line that is perpendicular to the line $y = 2x + 1$ and passes through the point $(3, 7)$, you need to follow these steps:

1. Find the slope of the given line. The slope of the given line is 2.
2. Find the negative reciprocal of the slope of the given line. The negative reciprocal of 2 is $-\frac{1}{2}$.
3. Use the point-slope form of a line to write the equation of the perpendicular line.

$$y - 7 = -\frac{1}{2}(x - 3)$$

Simplifying the equation, you get:

$$y = -\frac{1}{2}x + 8$$

Q: What is the equation of a line that is perpendicular to the line $x = 4$ and passes through the point $(2, -3)$? A: To find the equation of a line that is perpendicular to the line $x = 4$ and passes through the point $(2, -3)$, you need to follow these steps:

1. Find the equation of the vertical line. The equation of the vertical line is $x = 4$.
2. The equation of the perpendicular line will be a horizontal line. It can be represented by the equation:

$$y = a$$

where $a$ is a constant.

Substituting the coordinates of the point into the equation, you get:

$$-3 = a$$

Therefore, the equation of the perpendicular line is:

$$y = -3$$

Conclusion

In conclusion, the concept of perpendicular lines and slope is essential in determining the equation of a line. The equation of the perpendicular line is found by substituting the coordinates of the point into the equation. The slope of the given line is used to determine the slope of the perpendicular line, which is the negative reciprocal of the slope of the given line.