Conic Sections

• Introduction

  • Conic Sections definition
  • Formation by intersecting a right circular cone and a plane
  • Types of sections based on angle ($\alpha$) and slope ($\beta$):
    • Circle ($\beta = 90^\circ$)
    • Ellipse ($\alpha < \beta < 90^\circ$)
    • Parabola ($\beta = \alpha$)
    • Hyperbola ($\beta < \alpha$)
  • Degenerate conics (Point, Line, Pair of intersecting lines)

• Circles

  • Definition: locus of a point moving at a constant distance (radius) from a fixed point (center)
  • Equation forms:
    • Standard form: $(x - h)^2 + (y - k)^2 = r^2$
    • Origin center: $x^2 + y^2 = r^2$
    • General form: $x^2 + y^2 + 2gx + 2fy + c = 0$

• Parabola

  • Definition: locus of a point equidistant from a fixed point (focus) and a fixed line (directrix)
  • Key terms: Axis, Vertex, Latus rectum
  • Four standard equations:
    • Right-opening: $y^2 = 4ax$
    • Left-opening: $y^2 = -4ax$
    • Upward-opening: $x^2 = 4ay$
    • Downward-opening: $x^2 = -4ay$
  • Key parameters: Focus coordinates, Directrix equation, Length of latus rectum ($4a$)

• Ellipse

  • Definition: locus of a point moving such that the sum of its distances from two fixed points (foci) is constant ($2a$)
  • Key terms: Major axis, Minor axis, Center, Vertices, Eccentricity ($e < 1$)
  • Standard equations:
    • Horizontal major axis: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b$)
    • Vertical major axis: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ ($a > b$)
  • Key relations: $c^2 = a^2 - b^2$, $e = \frac{c}{a}$, Latus rectum = $\frac{2b^2}{a}$

• Hyperbola

  • Definition: locus of a point where the difference of its distances from two fixed points (foci) is constant ($2a$)
  • Key terms: Transverse axis, Conjugate axis, Vertices, Eccentricity ($e > 1$)
  • Standard equations:
    • Horizontal transverse axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
    • Vertical transverse axis: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
  • Key relations: $c^2 = a^2 + b^2$, $e = \frac{c}{a}$, Latus rectum = $\frac{2b^2}{a}$
  • Asymptotes (lines approaching the curve)