Finding the Slope of the Given Lines

Oct 13, 2025 by talynt.org 111 views

Finding the Equation of a Pair of Lines Passing Through a Given Point and Parallel to Another Pair of Lines

In mathematics, finding the equation of a pair of lines that pass through a given point and are parallel to another pair of lines is a fundamental problem in algebra and geometry. This problem involves understanding the properties of lines, including their equations, slopes, and points of intersection. In this article, we will explore the steps to find the equation of a pair of lines passing through the point ${$ (1,0)$}$ and parallel to the lines represented by the equation ${ $3x^2 + xy - 10y^2 = 0\$}$ .

The given equation ${ $3x^2 + xy - 10y^2 = 0\$}$ represents a pair of lines in the Cartesian plane. To find the equation of a pair of lines parallel to this pair, we need to understand the properties of the given equation. The equation can be rewritten in the form ${$ Ax^2 + Bxy + Cy^2 = 0$}$, where ${$ A = 3$}$, ${$ B = 1$}$, and ${$ C = -10$}$.

Finding the Slope of the Given Lines

The slope of a line represented by the equation ${$ Ax^2 + Bxy + Cy^2 = 0$}$ can be found using the formula ${$ m = -\frac{B}{2A}$}$. In this case, the slope of the given lines is ${$ m = -\frac{1}{2(3)} = -\frac{1}{6}$}$.

Finding the Equation of the Parallel Lines

Since the lines we want to find are parallel to the given lines, they will have the same slope, which is ${$ -\frac{1}{6}$}$. To find the equation of these lines, we need to use the point-slope form of a line, which is ${$ y - y_1 = m(x - x_1)$}$, where ${$ m$}$ is the slope and ${$ (x_1, y_1)$}$ is a point on the line.

Using the Point-Slope Form

Substituting the slope ${$ m = -\frac{1}{6}$}$ and the point ${$ (1,0)$}$ into the point-slope form, we get ${$ y - 0 = -\frac{1}{6}(x - 1)$}$. Simplifying this equation, we get ${$ y = -\frac{1}{6}x + \frac{1}{6}$}$.

Finding the Equation of the Second Line

Since the lines we want to find are a pair of lines, they will have the same slope but different y-intercepts. To find the equation of the second line, we can use the fact that the two lines are symmetric about the x-axis. This means that the y-intercept of the second line will be the negative of the y-intercept of the first line.

Finding the Y-Intercept of the Second Line

The y-intercept of the first line is ${$ \frac{1}{6}$}$. Therefore, the y-intercept of the second line is ${$ -\frac{1}{6}$}$.

Finding the Equation of the Second Line

Using the point-slope form again, we can find the equation of the second line. Substituting the slope ${$ m = -\frac{1}{6}$}$ and the y-intercept ${$ -\frac{1}{6}$}$ into the point-slope form, we get ${$ y - (-\frac{1}{6}) = -\frac{1}{6}(x - 1)$}$. Simplifying this equation, we get ${$ y + \frac{1}{6} = -\frac{1}{6}x + \frac{1}{6}$}$. Combining like terms, we get ${$ y = -\frac{1}{6}x$}$.

Finding the Equation of the Pair of Lines

To find the equation of the pair of lines, we need to combine the equations of the two lines. We can do this by adding the two equations together. The equation of the pair of lines is ${ $3x^2 + xy - 10y^2 - 6x - y + 3 = 0\$}$ .

In this article, we have shown how to find the equation of a pair of lines passing through a given point and parallel to another pair of lines. We have used the point-slope form of a line and the properties of the given equation to find the equation of the pair of lines. The equation of the pair of lines is ${ $3x^2 + xy - 10y^2 - 6x - y + 3 = 0\$}$ . This problem is an important application of algebra and geometry, and it has many practical uses in fields such as engineering and physics.
Q&A: Finding the Equation of a Pair of Lines Passing Through a Given Point and Parallel to Another Pair of Lines

A: The main concept behind finding the equation of a pair of lines passing through a given point and parallel to another pair of lines is to use the properties of the given equation and the point-slope form of a line. The given equation represents a pair of lines, and we need to find the equation of a pair of lines that pass through a given point and are parallel to the given lines.

Q: How do we find the slope of the given lines?

A: To find the slope of the given lines, we use the formula ${$ m = -\frac{B}{2A}$}$, where ${$ A$}$ and ${$ B$}$ are coefficients of the given equation. In this case, the slope of the given lines is ${$ m = -\frac{1}{2(3)} = -\frac{1}{6}$}$.

Q: How do we find the equation of the parallel lines?

A: To find the equation of the parallel lines, we use the point-slope form of a line, which is ${$ y - y_1 = m(x - x_1)$}$, where ${$ m$}$ is the slope and ${$ (x_1, y_1)$}$ is a point on the line. We substitute the slope ${$ m = -\frac{1}{6}$}$ and the point ${$ (1,0)$}$ into the point-slope form to get the equation of the parallel lines.

Q: How do we find the equation of the second line?

A: To find the equation of the second line, we use the fact that the two lines are symmetric about the x-axis. This means that the y-intercept of the second line will be the negative of the y-intercept of the first line. We then use the point-slope form again to find the equation of the second line.

Q: What is the equation of the pair of lines?

A: The equation of the pair of lines is ${ $3x^2 + xy - 10y^2 - 6x - y + 3 = 0\$}$ .

A: Some practical applications of finding the equation of a pair of lines passing through a given point and parallel to another pair of lines include:

Engineering: Finding the equation of a pair of lines can be used to design and analyze mechanical systems, such as gears and linkages.
Physics: Finding the equation of a pair of lines can be used to model and analyze the motion of objects, such as projectiles and pendulums.
Computer Science: Finding the equation of a pair of lines can be used to develop algorithms for computer graphics and game development.

A: Some common mistakes to avoid when finding the equation of a pair of lines passing through a given point and parallel to another pair of lines include:

Failing to use the point-slope form of a line.
Failing to use the properties of the given equation.
Making errors in the calculations.

A: You can practice finding the equation of a pair of lines passing through a given point and parallel to another pair of lines by:

Working through examples and exercises.
Using online resources and tutorials.
Practicing with different types of problems and equations.

In this article, we have provided a Q&A section to help you understand the concept of finding the equation of a pair of lines passing through a given point and parallel to another pair of lines. We have covered topics such as the main concept behind finding the equation of a pair of lines, finding the slope of the given lines, finding the equation of the parallel lines, and some practical applications of finding the equation of a pair of lines. We hope this article has been helpful in your understanding of this concept.