1. What is a Rotated Conic?

A rotated conic is a conic section (ellipse, parabola, or hyperbola) that has been rotated about the origin by a specific angle. The rotation transforms the standard conic equation by introducing an xy term.

2. How Does the Calculator Work?

The calculator uses rotation transformation formulas:

Where:

• \( (x, y) \) — Original coordinates
• \( (x', y') \) — Rotated coordinates
• \( \theta \) — Rotation angle (degrees)

Explanation: The calculator substitutes the rotation formulas into the original conic equation and simplifies to obtain the transformed equation in the new coordinate system.

3. Importance of Conic Rotation

Details: Rotating conics helps eliminate the xy term, making it easier to identify and analyze the type of conic section. This is particularly useful in geometry, physics, and engineering applications.

4. Using the Calculator

Tips: Enter all six coefficients (A-F) of the original conic equation and the rotation angle in degrees. The calculator will compute and display the transformed equation after rotation.

5. Frequently Asked Questions (FAQ)

Q1: What types of conics can be rotated?
A: All standard conic sections - ellipses, parabolas, and hyperbolas - can be rotated using this method.

Q2: Why rotate conic sections?
A: Rotation helps eliminate the cross-term (xy) which simplifies the equation and makes it easier to identify the conic type and its properties.

Q3: What is the range for rotation angle?
A: The rotation angle can be any real number, though angles between 0° and 360° are most commonly used.

Q4: Does rotation change the conic type?
A: No, rotation is a rigid transformation that preserves the fundamental properties and type of the conic section.

Q5: Can I rotate conics about points other than the origin?
A: This calculator rotates about the origin. For rotation about other points, translation to the origin is required first.