Find Equation Of Line Through (4) (-2,-6) And Perpendicular To The Line 6x+y-14=0 TR
Introduction
In mathematics, finding the equation of a line that passes through a given point and is perpendicular to another line is a fundamental problem in geometry and algebra. This problem involves understanding the concept of slope and perpendicular lines. In this article, we will discuss how to find the equation of a line that passes through the point (4, -2, -6) and is perpendicular to the line 6x + y - 14 = 0.
Understanding the Problem
To find the equation of a line that passes through a given point and is perpendicular to another line, we need to understand the concept of slope and perpendicular lines. The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Two lines are perpendicular if their slopes are negative reciprocals of each other.
Finding the Slope of the Given Line
The given line is 6x + y - 14 = 0. To find its slope, we need to rewrite it in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. We can rewrite the given line as follows:
y = -6x + 14
From this equation, we can see that the slope of the given line is -6.
Finding the Slope of the Perpendicular Line
Since the line we are looking for is perpendicular to the given line, its slope will be the negative reciprocal of the slope of the given line. The negative reciprocal of -6 is 1/6.
Finding the Equation of the Perpendicular Line
Now that we have the slope of the perpendicular line, we can use the point-slope form of a line to find its equation. The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope. Plugging in the values, we get:
y - (-2) = (1/6)(x - 4)
Simplifying this equation, we get:
y + 2 = (1/6)(x - 4)
Multiplying both sides by 6 to eliminate the fraction, we get:
6(y + 2) = x - 4
Expanding the left-hand side, we get:
6y + 12 = x - 4
Subtracting 12 from both sides, we get:
6y = x - 16
Dividing both sides by 6, we get:
y = (1/6)x - 8/3
Conclusion
In this article, we discussed how to find the equation of a line that passes through a given point and is perpendicular to another line. We used the concept of slope and perpendicular lines to find the equation of the perpendicular line. The equation of the perpendicular line is y = (1/6)x - 8/3.
Example Problems
Problem 1
Find the equation of a line that passes through the point (2, 3) and is perpendicular to the line x + 2y - 5 = 0.
Solution
To find the equation of the perpendicular line, we need to find the slope of the given line. The given line is x + 2y - 5 = 0. We can rewrite it in the slope-intercept form as follows:
y = (-1/2)x + 5/2
The slope of the given line is -1/2. The negative reciprocal of -1/2 is 2. Now, we can use the point-slope form to find the equation of the perpendicular line:
y - 3 = 2(x - 2)
Simplifying this equation, we get:
y - 3 = 2x - 4
Adding 3 to both sides, we get:
y = 2x - 1
Problem 2
Find the equation of a line that passes through the point (1, 2) and is perpendicular to the line 3x + 2y - 7 = 0.
Solution
To find the equation of the perpendicular line, we need to find the slope of the given line. The given line is 3x + 2y - 7 = 0. We can rewrite it in the slope-intercept form as follows:
y = (-3/2)x + 7/2
The slope of the given line is -3/2. The negative reciprocal of -3/2 is 2/3. Now, we can use the point-slope form to find the equation of the perpendicular line:
y - 2 = (2/3)(x - 1)
Simplifying this equation, we get:
y - 2 = (2/3)x - 2/3
Adding 2 to both sides, we get:
y = (2/3)x + 4/3
Applications of Perpendicular Lines
Perpendicular lines have many applications in real-life situations. Some of the applications include:
- Geometry: Perpendicular lines are used to find the length of a line segment, the area of a triangle, and the volume of a pyramid.
- Physics: Perpendicular lines are used to describe the motion of objects, such as the trajectory of a projectile or the path of a rolling ball.
- Engineering: Perpendicular lines are used to design buildings, bridges, and other structures.
- Computer Science: Perpendicular lines are used in computer graphics to create 3D models and animations.
Conclusion
In this article, we discussed how to find the equation of a line that passes through a given point and is perpendicular to another line. We used the concept of slope and perpendicular lines to find the equation of the perpendicular line. The equation of the perpendicular line is y = (1/6)x - 8/3. We also discussed some example problems and applications of perpendicular lines.
Introduction
In our previous article, we discussed how to find the equation of a line that passes through a given point and is perpendicular to another line. In this article, we will answer some frequently asked questions related to this topic.
Q&A
Q1: What is the slope of a line that is perpendicular to the line 2x + 3y - 5 = 0?
A1: To find the slope of a line that is perpendicular to the given line, we need to find the slope of the given line. The given line is 2x + 3y - 5 = 0. We can rewrite it in the slope-intercept form as follows:
y = (-2/3)x + 5/3
The slope of the given line is -2/3. The negative reciprocal of -2/3 is 3/2.
Q2: How do I find the equation of a line that passes through the point (1, 2) and is perpendicular to the line x + 2y - 3 = 0?
A2: To find the equation of the perpendicular line, we need to find the slope of the given line. The given line is x + 2y - 3 = 0. We can rewrite it in the slope-intercept form as follows:
y = (-1/2)x + 3/2
The slope of the given line is -1/2. The negative reciprocal of -1/2 is 2. Now, we can use the point-slope form to find the equation of the perpendicular line:
y - 2 = 2(x - 1)
Simplifying this equation, we get:
y - 2 = 2x - 2
Adding 2 to both sides, we get:
y = 2x
Q3: What is the equation of a line that passes through the point (3, 4) and is perpendicular to the line 4x + 2y - 10 = 0?
A3: To find the equation of the perpendicular line, we need to find the slope of the given line. The given line is 4x + 2y - 10 = 0. We can rewrite it in the slope-intercept form as follows:
y = (-2)x + 5
The slope of the given line is -2. The negative reciprocal of -2 is 1/2. Now, we can use the point-slope form to find the equation of the perpendicular line:
y - 4 = (1/2)(x - 3)
Simplifying this equation, we get:
y - 4 = (1/2)x - 3/2
Adding 4 to both sides, we get:
y = (1/2)x + 5/2
Q4: How do I find the equation of a line that passes through the point (2, 1) and is perpendicular to the line 3x + 4y - 7 = 0?
A4: To find the equation of the perpendicular line, we need to find the slope of the given line. The given line is 3x + 4y - 7 = 0. We can rewrite it in the slope-intercept form as follows:
y = (-3/4)x + 7/4
The slope of the given line is -3/4. The negative reciprocal of -3/4 is 4/3. Now, we can use the point-slope form to find the equation of the perpendicular line:
y - 1 = (4/3)(x - 2)
Simplifying this equation, we get:
y - 1 = (4/3)x - 8/3
Adding 1 to both sides, we get:
y = (4/3)x - 5/3
Conclusion
In this article, we answered some frequently asked questions related to finding the equation of a line that passes through a given point and is perpendicular to another line. We hope that this article has been helpful in clarifying any doubts you may have had.
Example Problems
Problem 1
Find the equation of a line that passes through the point (1, 2) and is perpendicular to the line x + 2y - 3 = 0.
Solution
To find the equation of the perpendicular line, we need to find the slope of the given line. The given line is x + 2y - 3 = 0. We can rewrite it in the slope-intercept form as follows:
y = (-1/2)x + 3/2
The slope of the given line is -1/2. The negative reciprocal of -1/2 is 2. Now, we can use the point-slope form to find the equation of the perpendicular line:
y - 2 = 2(x - 1)
Simplifying this equation, we get:
y - 2 = 2x - 2
Adding 2 to both sides, we get:
y = 2x
Problem 2
Find the equation of a line that passes through the point (3, 4) and is perpendicular to the line 4x + 2y - 10 = 0.
Solution
To find the equation of the perpendicular line, we need to find the slope of the given line. The given line is 4x + 2y - 10 = 0. We can rewrite it in the slope-intercept form as follows:
y = (-2)x + 5
The slope of the given line is -2. The negative reciprocal of -2 is 1/2. Now, we can use the point-slope form to find the equation of the perpendicular line:
y - 4 = (1/2)(x - 3)
Simplifying this equation, we get:
y - 4 = (1/2)x - 3/2
Adding 4 to both sides, we get:
y = (1/2)x + 5/2
Applications of Perpendicular Lines
Perpendicular lines have many applications in real-life situations. Some of the applications include:
- Geometry: Perpendicular lines are used to find the length of a line segment, the area of a triangle, and the volume of a pyramid.
- Physics: Perpendicular lines are used to describe the motion of objects, such as the trajectory of a projectile or the path of a rolling ball.
- Engineering: Perpendicular lines are used to design buildings, bridges, and other structures.
- Computer Science: Perpendicular lines are used in computer graphics to create 3D models and animations.
Conclusion
In this article, we answered some frequently asked questions related to finding the equation of a line that passes through a given point and is perpendicular to another line. We hope that this article has been helpful in clarifying any doubts you may have had.