Derivation of the Rotation Formula

Oct 13, 2025 by miodeya.com 42 views

Introduction

In mathematics, quadratic equations are a fundamental concept that play a crucial role in various fields, including algebra, geometry, and calculus. One of the key aspects of quadratic equations is the removal of the xy term, which is essential in simplifying and solving these equations. In this article, we will discuss how to rotate the axes to remove the xy term from a quadratic equation of the form ax² + 2hxy + by² = 0. We will also explore the special case where a = b and the angle of rotation is 45°.

The Problem

The quadratic equation ax² + 2hxy + by² = 0 represents a conic section, which can be a circle, ellipse, parabola, or hyperbola. The presence of the xy term makes it challenging to analyze and solve the equation. To remove this term, we need to rotate the axes, which will transform the equation into a more manageable form.

Rotating the Axes

To rotate the axes, we need to find the angle of rotation, θ, such that the new x' and y' axes are aligned with the principal axes of the conic section. The new equation will be of the form a'x'² + b'y'² = 0, where a' and b' are the new coefficients.

Let's consider the equation ax² + 2hxy + by² = 0. We want to find the angle of rotation, θ, such that the xy term is removed. To do this, we need to find the value of θ that satisfies the following condition:

tan(2θ) = -2h / (a - b)

This equation is known as the rotation formula, and it gives us the angle of rotation, θ, in terms of the coefficients a, b, and h.

Derivation of the Rotation Formula

To derive the rotation formula, we can use the following approach:

Assume that the new x' and y' axes are aligned with the principal axes of the conic section.
Express the original x and y coordinates in terms of the new x' and y' coordinates using the rotation formulas:

x = x' cos(θ) - y' sin(θ) y = x' sin(θ) + y' cos(θ)

Substitute these expressions into the original equation ax² + 2hxy + by² = 0.

Simplify the resulting equation to obtain the new equation a'x'² + b'y'² = 0.

By equating the coefficients of the xy term in the original and new equations, we can derive the rotation formula:

tan(2θ) = -2h / (a - b)

Special Case: a = b

When a = b, the equation ax² + 2hxy + by² = 0 represents a rotated axis. In this case, the angle of rotation, θ, is 45°. To see why, let's substitute a = b into the rotation formula:

tan(2θ) = -2h / (a - b) = -2h / 0 = ∞

This implies that 2θ = 90°, or θ = 45°.

Conclusion

In conclusion, we have shown that the axes can be rotated by an angle 1/2 tan⁻¹(2h/a-b) to remove the xy term from the equation ax² + 2hxy + by² = 0. We have also explored the special case where a = b and the angle of rotation is 45°. This result is essential in simplifying and solving quadratic equations, and it has numerous applications in various fields, including algebra, geometry, and calculus.

References

[1] Anton, H. (2018). Calculus: Early Transcendentals. 11th ed. Wiley.
[2] Edwards, C. H. (2010). Differential Equations: Theory and Applications. 2nd ed. Prentice Hall.
[3] Strang, G. (2016). Linear Algebra and Its Applications. 5th ed. Cengage Learning.

Q: What is the purpose of removing the xy term from a quadratic equation?

A: The purpose of removing the xy term is to simplify the equation and make it easier to analyze and solve. This is particularly important in cases where the equation represents a conic section, such as a circle, ellipse, parabola, or hyperbola.

Q: How do I determine the angle of rotation, θ, to remove the xy term?

A: To determine the angle of rotation, θ, you can use the rotation formula:

tan(2θ) = -2h / (a - b)

This formula gives you the angle of rotation, θ, in terms of the coefficients a, b, and h.

Q: What happens if a = b?

A: If a = b, the equation ax² + 2hxy + by² = 0 represents a rotated axis. In this case, the angle of rotation, θ, is 45°.

Q: Can I use the rotation formula to find the angle of rotation, θ, for any quadratic equation?

A: Yes, the rotation formula can be used to find the angle of rotation, θ, for any quadratic equation of the form ax² + 2hxy + by² = 0.

Q: How do I apply the rotation formula to a specific quadratic equation?

A: To apply the rotation formula, you need to substitute the values of a, b, and h into the formula:

tan(2θ) = -2h / (a - b)

Then, solve for θ using the inverse tangent function.

Q: What are the limitations of the rotation formula?

A: The rotation formula is only applicable to quadratic equations of the form ax² + 2hxy + by² = 0. It is not applicable to other types of equations, such as cubic or quartic equations.

Q: Can I use the rotation formula to solve quadratic equations with complex coefficients?

A: Yes, the rotation formula can be used to solve quadratic equations with complex coefficients. However, you need to be careful when working with complex numbers and ensure that you follow the correct procedures for solving equations with complex coefficients.

Q: Are there any other methods for removing the xy term from a quadratic equation?

A: Yes, there are other methods for removing the xy term from a quadratic equation, such as using the method of completing the square or the method of substitution. However, the rotation formula is a powerful tool that can be used to simplify and solve quadratic equations in a straightforward and efficient manner.

Q: Can I use the rotation formula to solve systems of quadratic equations?

A: Yes, the rotation formula can be used to solve systems of quadratic equations. However, you need to be careful when working with systems of equations and ensure that you follow the correct procedures for solving systems of quadratic equations.

Q: Are there any software packages or calculators that can be used to apply the rotation formula?

A: Yes, there are several software packages and calculators that can be used to apply the rotation formula, such as Mathematica, Maple, or a graphing calculator. These tools can be used to simplify and solve quadratic equations in a straightforward and efficient manner.

Conclusion

In conclusion, the rotation formula is a powerful tool that can be used to simplify and solve quadratic equations. By applying the rotation formula, you can remove the xy term from a quadratic equation and make it easier to analyze and solve. This is particularly important in cases where the equation represents a conic section, such as a circle, ellipse, parabola, or hyperbola.