12.4 Rotation of Axes

Conic sections are fascinating curves formed by slicing a cone. They include circles, ellipses, parabolas, and hyperbolas. Each has unique properties and can be described by specific equations, which we'll learn to identify and manipulate.

Understanding conic sections is crucial for many real-world applications. From satellite orbits to architectural design, these curves play a vital role. We'll explore how to analyze and transform them, unlocking their potential in various fields.

Conic Sections

General form of conic sections

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• General form of a conic section $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$
  • Coefficients $A$, $B$, and $C$ cannot all be zero at the same time
• Conic sections classified based on the discriminant $B^2 - 4AC$
  • Ellipse when $B^2 - 4AC < 0$ (circle, oval)
  • Parabola when $B^2 - 4AC = 0$ (U-shaped curve)
  • Hyperbola when $B^2 - 4AC > 0$ (two separate curved parts)
• Circle is a special case of an ellipse where $B = 0$ and $A = C$

Rotation of axes for conics

• Rotation of axes eliminates the $xy$ term in the general form equation
• Angle of rotation $\theta$ calculated using $\tan 2\theta = \frac{B}{A-C}$
  • Choose angle between $0^\circ$ and $90^\circ$
• Substitution formulas for rotation of axes:
  • $x = x'\cos\theta - y'\sin\theta$
  • $y = x'\sin\theta + y'\cos\theta$
• Substitute formulas, expand, and simplify to obtain rotated form
• Trigonometry is essential for calculating the rotation angle and performing the axis transformation

Standard form of rotated conics

• Rotated form of a conic section $A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0$
• Complete the square for both $x'$ and $y'$ terms:
  1. Divide equation by constant term of $x'^2$ (or $y'^2$ if $x'^2$ term missing)
  2. Move constant terms to right side of equation
  3. Factor out coefficients of $x'$ and $y'$
  4. Add square of half the coefficient of $x'$ (or $y'$) to both sides
• Rewrite equation in standard form by renaming variables and simplifying

Analysis of non-rotated conics

• Identify conic section type using discriminant $B^2 - 4AC$
• For ellipses and hyperbolas:
  • Center $(h, k)$ found by completing square for $x$ and $y$ terms
  • Lengths of major and minor axes determined using coefficients of $x^2$ and $y^2$
  • Eccentricity calculated to measure deviation from a circle (0 for circle, between 0 and 1 for ellipse, greater than 1 for hyperbola)
• For parabolas:
  • Vertex identified by completing square for variable with squared term
  • Direction of opening based on sign of coefficient of squared term (positive opens upward, negative opens downward)
  • Focus and directrix found using standard form equation

Mathematical Foundations

• Coordinate plane: The 2D system where conic sections are graphed and analyzed
• Quadratic equations: Form the basis for describing conic sections mathematically
• Linear algebra: Provides tools for transforming conic sections, including rotation and translation of axes