General Form of a Pair of Lines

Oct 13, 2025 by talynt.org 74 views

**Find the Equation of a Pair of Lines Passing Through a Given Point and Parallel to Another Pair of Lines** ===========================================================

Introduction

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 geometry and algebra. This problem involves understanding the concept of parallel lines, the general form of a pair of lines, and the process of finding the equation of a line passing through a given point.

Understanding the Problem

The problem requires finding the equation of a pair of lines that pass through the point (1, 0) and are parallel to the lines represented by the equation 3x^2 + xy - 10y^2 = 0. To solve this problem, we need to understand the concept of parallel lines and the general form of a pair of lines.

What are Parallel Lines?

Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. In other words, parallel lines are lines that have the same slope but different y-intercepts.

General Form of a Pair of Lines

A pair of lines can be represented in the general form:

ax^2 + bxy + cy^2 = 0

where a, b, and c are constants.

Finding the Equation of a Pair of Lines

To find the equation of a pair of lines that pass through a given point and are parallel to another pair of lines, we need to follow these steps:

Find the slope of the given pair of lines: The slope of the given pair of lines can be found by comparing the coefficients of the x^2 and y^2 terms in the equation 3x^2 + xy - 10y^2 = 0.
Find the equation of a line passing through the given point and having the same slope as the given pair of lines: We can use the point-slope form of a line to find the equation of a line passing through the point (1, 0) and having the same slope as the given pair of lines.
Find the equation of the other line: Since the two lines are parallel, they have the same slope. Therefore, we can find the equation of the other line by using the point-slope form of a line and the slope of the given pair of lines.

Step-by-Step Solution

Step 1: Find the Slope of the Given Pair of Lines

The slope of the given pair of lines can be found by comparing the coefficients of the x^2 and y^2 terms in the equation 3x^2 + xy - 10y^2 = 0.

import sympy as sp
x, y = sp.symbols('x y')
equation = 3x**2 + xy - 10*y**2
slope = sp.solve(equation, y)[0].coeff(x)
print(slope)

Step 2: Find the Equation of a Line Passing Through the Given Point and Having the Same Slope as the Given Pair of Lines

We can use the point-slope form of a line to find the equation of a line passing through the point (1, 0) and having the same slope as the given pair of lines.

# Define the point
point = (1, 0)
slope = -3/10
equation = sp.Eq(y - point[1], slope*(x - point[0]))
print(equation)

Step 3: Find the Equation of the Other Line

Since the two lines are parallel, they have the same slope. Therefore, we can find the equation of the other line by using the point-slope form of a line and the slope of the given pair of lines.

# Define the point
point = (1, 0)
slope = -3/10
equation = sp.Eq(y - point[1], slope*(x - point[0]))
print(equation)

Conclusion

In this article, we discussed how to find the equation of a pair of lines that pass through a given point and are parallel to another pair of lines. We used the point-slope form of a line and the concept of parallel lines to find the equation of a pair of lines. We also provided a step-by-step solution to the problem using Python code.

Q&A

Q: What is the equation of a pair of lines that pass through the point (1, 0) and are parallel to the lines represented by the equation 3x^2 + xy - 10y^2 = 0?

A: The equation of a pair of lines that pass through the point (1, 0) and are parallel to the lines represented by the equation 3x^2 + xy - 10y^2 = 0 is:

y - 0 = (-3/10)(x - 1)

Q: How do I find the equation of a pair of lines that pass through a given point and are parallel to another pair of lines?

A: To find the equation of a pair of lines that pass through a given point and are parallel to another pair of lines, you need to follow these steps:

Find the slope of the given pair of lines.
Find the equation of a line passing through the given point and having the same slope as the given pair of lines.
Find the equation of the other line.

Q: What is the slope of the given pair of lines?

A: The slope of the given pair of lines is -3/10.

Q: How do I find the equation of a line passing through a given point and having a given slope?

A: To find the equation of a line passing through a given point and having a given slope, you can use the point-slope form of a line:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the given slope.