Conic sections are fascinating curves formed by intersecting a plane with a double cone. They include circles, ellipses, parabolas, and hyperbolas. Each type has unique properties and equations that can be transformed through rotation and translation.

Understanding conic sections is crucial for grasping more advanced mathematical concepts. These curves appear in various real-world applications, from satellite orbits to architectural designs. Mastering their equations and transformations provides a solid foundation for further study in mathematics and physics.

Conic Sections and Rotation of Axes

General form of conic sections

• Conic sections represented by general form equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$
• Coefficients $A$, $B$, and $C$ determine the type of conic section
  • $B = 0$ indicates conic is not rotated
    • $A = C$ represents a circle
    • $A = 0$ or $C = 0$ represents a parabola
    • $A$ and $C$ having the same sign represents an ellipse (oval shape)
    • $A$ and $C$ having opposite signs represents a hyperbola (two separate curves)
  • $B \neq 0$ indicates conic is rotated at an angle to the coordinate axes

Rotation of axes for conics

• Rotation angle $\theta$ calculated using $\theta = \frac{1}{2} \arctan \left(\frac{B}{A-C}\right)$
• Rotation formulas transform coordinates from original $(x, y)$ to rotated $(x', y')$ system (coordinate transformation)
  • $x = x' \cos \theta - y' \sin \theta$ transforms $x$-coordinate
  • $y = x' \sin \theta + y' \cos \theta$ transforms $y$-coordinate
• Substituting rotation formulas into general form equation eliminates $xy$ term
  • Resulting equation in rotated system: $A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0$
  • Coefficients $A'$, $C'$, $D'$, $E'$, and $F'$ differ from original equation

Standard form of rotated conics

• After applying rotation of axes, conic equation simplified to $A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0$
• Completing the square for both $x'$ and $y'$ terms isolates squared terms
  • $A'(x' + \frac{D'}{2A'})^2 + C'(y' + \frac{E'}{2C'})^2 = -F' + \frac{D'^2}{4A'} + \frac{E'^2}{4C'}$
• Translating conic to center it at origin using $x'' = x' + \frac{D'}{2A'}$ and $y'' = y' + \frac{E'}{2C'}$
• Standard form equation depends on conic type
  • Ellipse: $\frac{x''^2}{a^2} + \frac{y''^2}{b^2} = 1$ (sum of squared terms equals 1)
  • Hyperbola: $\frac{x''^2}{a^2} - \frac{y''^2}{b^2} = 1$ (difference of squared terms equals 1)
  • Parabola: $y'' = ax''^2 + bx''$ or $x'' = ay''^2 + by''$ (single squared term)

Direct analysis of conic equations

• Conic type determined by coefficients in general form equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$
• Center, vertices, foci, and directrices found using formulas specific to each conic type
  • Ellipse:
    • Center located at $(-\frac{D}{2A}, -\frac{E}{2C})$
    • Vertices at $(\pm a, 0)$ and $(0, \pm b)$ after translation (endpoints of major and minor axes)
    • Foci at $(\pm c, 0)$ and $(0, \pm c)$ after translation, where $c^2 = a^2 - b^2$ (sum of distances to foci is constant)
  • Hyperbola:
    • Center located at $(-\frac{D}{2A}, -\frac{E}{2C})$
    • Vertices at $(\pm a, 0)$ after translation (endpoints of transverse axis)
    • Foci at $(\pm c, 0)$ after translation, where $c^2 = a^2 + b^2$ (difference of distances to foci is constant)
  • Parabola:
    • Vertex at $(-\frac{D}{2A}, -\frac{E}{4A})$ or $(-\frac{E}{4C}, -\frac{D}{2C})$ (turning point)
    • Focus at $(-\frac{D}{2A}, \frac{1}{4A})$ or $(\frac{1}{4C}, -\frac{D}{2C})$ (equidistant from vertex and directrix)
    • Directrix is line $y = -\frac{E}{4A}$ or $x = -\frac{D}{2C}$ (perpendicular to axis of symmetry)

Advanced Rotation Concepts

• Matrix rotation: Rotation of axes can be represented using 2x2 rotation matrices
• Quadratic forms: General form of conic sections can be expressed as quadratic forms in matrix notation
• Principal axes: Directions along which the conic section has maximum and minimum curvature, determined through eigenvalue analysis of the quadratic form