Problem 7 - Algebra and Trigonometry

Question

Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}-3 y^{2}=4, \quad \phi=60^{\circ}$$

Step-by-Step Solution

Answer

The equation in rotated coordinates is \(-X^2 - 2\sqrt{3}XY = 4\).

Step 1: Identifying Transformation Formulae

The transformation formulae for rotating a conic section are given by: \[ x = X \cos(\phi) - Y \sin(\phi) \] \[ y = X \sin(\phi) + Y \cos(\phi) \] where \( \phi = 60^{\circ} \) in this problem.

Step 2: Substitute Rotation Angle

Substitute \( \phi = 60^{\circ} \) into the transformation equations: - \( x = X \cdot \cos(60^{\circ}) - Y \cdot \sin(60^{\circ}) = \frac{1}{2}X - \frac{\sqrt{3}}{2}Y \)- \( y = X \cdot \sin(60^{\circ}) + Y \cdot \cos(60^{\circ}) = \frac{\sqrt{3}}{2}X + \frac{1}{2}Y \)

Step 3: Replace x and y in the Original Equation

Substitute the expressions for \(x\) and \(y\) into the original equation \(x^2 - 3y^2 = 4\):\[\left(\frac{1}{2}X - \frac{\sqrt{3}}{2}Y\right)^2 - 3\left(\frac{\sqrt{3}}{2}X + \frac{1}{2}Y \right)^2 = 4 \]

Step 4: Expand and Simplify the Expression

Expand the squared terms: \[\left(\frac{1}{2}X - \frac{\sqrt{3}}{2}Y\right)^2 = \frac{1}{4}X^2 - \frac{\sqrt{3}}{2}XY + \frac{3}{4}Y^2\]\[3\left(\frac{\sqrt{3}}{2}X + \frac{1}{2}Y \right)^2 = 3 \left( \frac{3}{4}X^2 + \frac{\sqrt{3}}{2}XY + \frac{1}{4}Y^2 \right) = \frac{9}{4}X^2 + \frac{3\sqrt{3}}{2}XY + \frac{3}{4}Y^2\]

Step 5: Combine Terms and Simplify the Equation

Combine like terms and simplify: \[\frac{1}{4}X^2 - \frac{\sqrt{3}}{2}XY + \frac{3}{4}Y^2 - \left( \frac{9}{4}X^2 + \frac{3\sqrt{3}}{2}XY + \frac{3}{4}Y^2 \right) = 4\]Simplifying, we have:\[-X^2 - 2\sqrt{3}XY = 4\]Thus, the transformed equation is:\[ -X^2 - 2\sqrt{3}XY = 4 \]

Key Concepts

Transformation Formulas

To understand how to rotate a conic section, we begin by using the transformation formulas. These formulas help us shift coordinates when the axes are rotated by an angle. For a given angle \( \phi \), the transformation uses trigonometric identities:

• \( x = X \cos(\phi) - Y \sin(\phi) \)
• \( y = X \sin(\phi) + Y \cos(\phi) \)

These equations express the old coordinates \( (x, y) \) in terms of new coordinates \( (X, Y) \) and the angle of rotation \( \phi \). By inserting the angle for this exercise, \( \phi = 60^{\circ} \), we compute:

• \( x = \frac{1}{2}X - \frac{\sqrt{3}}{2}Y \)
• \( y = \frac{\sqrt{3}}{2}X + \frac{1}{2}Y \)

These formulas are fundamental when re-expressing equations during rotation operations.

Conic Sections

Conic sections are curves obtained by intersecting a right circular cone with a plane. The main types include circles, ellipses, parabolas, and hyperbolas. Each conic section has a unique set of properties and equations. In this exercise, we are dealing with the equation \( x^2 - 3y^2 = 4 \), which resembles a hyperbola. Hyperbolas have two distinct parts, extending to infinity, displaying an open curve. They have the general form: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] For analyzing conic sections under rotation, understanding how each type behaves and their standard equations is crucial. This understanding aids in predicting changes when axes rotate and adapting the standard forms accordingly.

Angle of Rotation

The angle of rotation \( \phi \) affects how the coordinate plane is reoriented. It involves rotating the axes by a given angle to transform the shape's orientation. In the given problem, the angle is \(60^{\circ}\). This angle affects the transformation formulas, which change the xy-coordinates to new XY-coordinates. It also influences the look and position of the conic section on a graph. Rotation often simplifies complex equations by eliminating the cross-product term \(Bxy\) in the conic's general equation, making it easier to identify the type of conic. Practically, choosing an appropriate angle such as \(60^{\circ}\) serves to achieve a desired coordinate system. The trigonometric values for \(60^{\circ}\) (such as \( \cos(60^{\circ}) = \frac{1}{2} \) and \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \)) facilitate the conversion between coordinate pairs, enabling deeper understanding and effective solution of conic-based problems.