Understanding the Nature of the Given Equation

Oct 13, 2025 by windailys.com 109 views

Introduction

In mathematics, finding the equation of a line perpendicular to a given line is a fundamental concept in geometry and trigonometry. Given a quadratic equation representing a pair of lines, we can find the equation of a line perpendicular to these lines using various methods. In this article, we will explore one such method and prove a trigonometric identity related to the problem.

The Given Equation

The given equation is ${$ x^2 - xy - 2y^2 = 0$}$. This is a quadratic equation in two variables, x and y, representing a pair of lines. To find the equation of a line perpendicular to these lines, we need to first understand the nature of the given equation.

Understanding the Nature of the Given Equation

The given equation can be factored as ${$ (x - 2y)(x + y) = 0$}$. This means that the given equation represents two lines: ${$ x - 2y = 0$}$ and ${$ x + y = 0$}$.

Finding the Equation of a Line Perpendicular to the Given Lines

To find the equation of a line perpendicular to the given lines, we need to find the slope of the given lines. The slope of the line ${$ x - 2y = 0$}$ is 2, and the slope of the line ${$ x + y = 0$}$ is -1.

The Slope of a Perpendicular Line

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. Therefore, the slope of a line perpendicular to the line ${$ x - 2y = 0$}$ is -1/2, and the slope of a line perpendicular to the line ${$ x + y = 0$}$ is 1.

The Equation of a Line Perpendicular to the Given Lines

Using the point-slope form of a line, we can write the equation of a line perpendicular to the given lines as ${$ 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.

Proving the Trigonometric Identity

Now, let's prove the trigonometric identity [ $\cos^2 \theta + \sin^2 \theta \cdot \cos^2 \beta = \cos^2 \beta + \sin^2 \beta \cdot \cos^2 \beta\$}$ .

Using the Pythagorean Identity

We can start by using the Pythagorean identity ${$ \cos^2 \theta + \sin^2 \theta = 1$}$. Multiplying both sides of this equation by ${$ \cos^2 \beta$, we get [ $\cos^2 \theta \cdot \cos^2 \beta + \sin^2 \theta \cdot \cos^2 \beta = \cos^2 \beta\$}$ .

Using the Distributive Property

Using the distributive property, we can rewrite the left-hand side of the equation as ${$ \cos^2 \theta \cdot \cos^2 \beta + \sin^2 \theta \cdot \cos^2 \beta = \cos^2 \beta \cdot (\cos^2 \theta + \sin^2 \theta)$].

Simplifying the Equation

Using the Pythagorean identity, we can simplify the equation as [ $\cos^2 \beta \cdot (\cos^2 \theta + \sin^2 \theta) = \cos^2 \beta \cdot 1\$, which is equal to \[$ \cos^2 \beta$}$.

Conclusion

In this article, we found the equation of a line perpendicular to the given lines and proved a trigonometric identity related to the problem. The equation of a line perpendicular to the given lines 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. We also proved the trigonometric identity \[$ \cos^2 \theta + \sin^2 \theta \cdot \cos^2 \beta = \cos^2 \beta + \sin^2 \beta \cdot \cos^2 \beta$}$ using the Pythagorean identity and the distributive property.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Geometry: Seeing, Doing, Understanding" by Harold R. Jacobs

Code

import math

def find_perpendicular_line(x1, y1, m):
    # Find the equation of a line perpendicular to the given lines
    # using the point-slope form of a line
    return f"y - {y1} = {m}(x - {x1})"

# Test the function
print(find_perpendicular_line(1, 2, -1/2))

Note

Q: What is the given equation?

A: The given equation is ${$ x^2 - xy - 2y^2 = 0$}$. This is a quadratic equation in two variables, x and y, representing a pair of lines.

Q: How can we factor the given equation?

A: The given equation can be factored as ${$ (x - 2y)(x + y) = 0$}$. This means that the given equation represents two lines: ${$ x - 2y = 0$}$ and ${$ x + y = 0$}$.

Q: What is the slope of the given lines?

A: The slope of the line ${$ x - 2y = 0$}$ is 2, and the slope of the line ${$ x + y = 0$}$ is -1.

Q: How can we find the equation of a line perpendicular to the given lines?

A: To find the equation of a line perpendicular to the given lines, we need to find the slope of the given lines. The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. Therefore, the slope of a line perpendicular to the line ${$ x - 2y = 0$}$ is -1/2, and the slope of a line perpendicular to the line ${$ x + y = 0$}$ is 1.

Q: What is the equation of a line perpendicular to the given lines?

A: Using the point-slope form of a line, we can write the equation of a line perpendicular to the given lines as ${$ 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 can we prove the trigonometric identity?

A: We can start by using the Pythagorean identity [ $\cos^2 \theta + \sin^2 \theta = 1\$}$ . Multiplying both sides of this equation by ${$ \cos^2 \beta$, we get [ $\cos^2 \theta \cdot \cos^2 \beta + \sin^2 \theta \cdot \cos^2 \beta = \cos^2 \beta\$}$ .

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any numbers a, b, and c, ${$ a(b + c) = ab + ac$}$.

Q: How can we simplify the equation?

A: Using the Pythagorean identity, we can simplify the equation as ${$ \cos^2 \beta \cdot (\cos^2 \theta + \sin^2 \theta) = \cos^2 \beta \cdot 1$, which is equal to [ $\cos^2 \beta\$}$ .

Q: What is the conclusion of the article?

A: In this article, we found the equation of a line perpendicular to the given lines and proved a trigonometric identity related to the problem. The equation of a line perpendicular to the given lines 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. We also proved the trigonometric identity \[$ \cos^2 \theta + \sin^2 \theta \cdot \cos^2 \beta = \cos^2 \beta + \sin^2 \beta \cdot \cos^2 \beta$}$ using the Pythagorean identity and the distributive property.

Q: What are the references for the article?

A: The references for the article are:

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Geometry: Seeing, Doing, Understanding" by Harold R. Jacobs

Q: What are the further readings for the article?

A: The further readings for the article are:

[1] "Calculus" by Michael Spivak
[2] "Linear Algebra and Its Applications" by Gilbert Strang

Q: What is the code for the article?

A: The code for the article is a simple implementation of the point-slope form of a line and is not intended to be used for actual calculations. It is provided for illustrative purposes only.

import math

def find_perpendicular_line(x1, y1, m):
    # Find the equation of a line perpendicular to the given lines
    # using the point-slope form of a line
    return f"y - {y1} = {m}(x - {x1})"

# Test the function
print(find_perpendicular_line(1, 2, -1/2))

Note

This code is a simple implementation of the point-slope form of a line and is not intended to be used for actual calculations. It is provided for illustrative purposes only.