Problem 7 - Precalculus: Mathematical for Calculus

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 rotated conic equation becomes: \(X^2 + XY - Y^2 = 4\).

Step 1: Understanding the Rotation

We need to find the new equation of the given conic, which is in its original form as \(x^{2}-3 y^{2}=4\), after the coordinate axes have been rotated by an angle of \(\phi = 60^{\circ}\). This means we are applying a transformation to both \(x\) and \(y\) to find \(X\) and \(Y\).

Step 2: Using the Rotation Formulas

The formulas for rotating the coordinate system by an angle \(\phi\) are: \(x = X \cos(\phi) - Y \sin(\phi)\) and \(y = X \sin(\phi) + Y \cos(\phi)\). For \(\phi = 60^{\circ}\), \(\cos(60^{\circ}) = \frac{1}{2}\) and \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\).

Step 3: Express Original Variables in Terms of Rotated Variables

Substitute \(x = \frac{1}{2}X - \frac{\sqrt{3}}{2}Y\) and \(y = \frac{\sqrt{3}}{2}X + \frac{1}{2}Y\) into the original equation \(x^{2} - 3y^{2} = 4\).

Step 4: Substitute and Simplify

We substitute into the equation: \((\frac{1}{2}X - \frac{\sqrt{3}}{2}Y)^2 - 3(\frac{\sqrt{3}}{2}X + \frac{1}{2}Y)^2 = 4\). Expanding these squares and simplifying will give us the equation in \(X\) and \(Y\).

Step 5: Perform Necessary Algebra

After expanding and simplifying, we obtain equations for \(x^2\) and \(3y^2\). Combine like terms to find the new coefficients in terms of \(X\) and \(Y\). This will yield the final form of the equation in rotated axes.

Step 6: Write the New Conic Equation

After simplifying, the new equation, which is now dependent on \(X\) and \(Y\) will give us the rotated conic. The algebra results in a different expression compared to the original conic equation.

Key Concepts

Coordinate Transformation

A coordinate transformation is a process that redefines a geometric figure in a different coordinate system. In the context of conic sections, this often involves changing from the original coordinate system, represented by variables \(x\) and \(y\), to a rotated system, represented by new variables \(X\) and \(Y\). To achieve this transformation, a rotation is applied to the entire coordinate system, aligning the axes to a new orientation without altering the ultimate shape and properties of the conic section itself.

When a conic section such as \(x^2 - 3y^2 = 4\) undergoes a coordinate transformation because of a rotation, it enables us to analyze the conic's properties more easily in scenarios where its axes might not initially align with the standard coordinate axes. The transformation provides a system that's more aligned with the inherent symmetries and orientation of the given geometric figure.

• Coordinate transformation allows new analyses and interpretations of geometric figures by providing a different perspective.
• Essential for problems involving non-standard orientations where the axes are not simply aligned with horizontal and vertical directions.

Rotation of Axes

The rotation of axes is a geometric technique used to reposition a conic section's equation in a newly rotated coordinate plane. This technique rotates the original coordinate system by a specific angle, denoted by \(\phi\), into a new coordinate system. The axes themselves remain perpendicular, but they are effectively 'turned' within the plane.

For example, a rotation by \(60^\circ\) involves using the rotation formulas.- The formula for the new \(X\) coordinate is: \(x = X \cos(\phi) - Y \sin(\phi)\). - For the new \(Y\) coordinate: \(y = X \sin(\phi) + Y \cos(\phi)\).
By substituting the respective trigonometric values for \( \cos(60^\circ) = \frac{1}{2} \) and \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), the equation of the rotated axes is established.

• This helps transform the conic's equation into the \(XY\) plane, simplifying further analysis.
• The axes rotation is not a physical movement, but rather a mathematical redefinition of the coordinate system.

Conic Equation

A conic equation is a formula that represents a conic section, which can be an ellipse, parabola, or hyperbola. In our exercise, the conic equation is given as \(x^2 - 3y^2 = 4\), and after rotation, we aim to convert it to new terms \(X\) and \(Y\).

The unique aspect of conic equations under rotation is that while the appearance of the equation changes, the fundamental properties of the conic—such as focus and directrix—remain constant. The primary goal of determining a rotated conic equation is to obtain a form where the new axes are either aligned with or perpendicular to the conic's principal axes.

• The rotation transforms the equation's appearance but retains the conic's size, shape, and set position.
• Solving the rotated conic equation involves algebraic techniques to simplify and express the equation using rotated variables.

When solving problems related to rotated conic sections, being adept at manipulating conic equations helps you understand how geometric figures can behave under transformations and rotations within the coordinate plane.