Analytic Geometry - Equations of Ellipses

Analytic Geometry - Equations of Ellipses

• Ellipse definition: A conic section formed by the set of points where the sum of distances to two foci is constant.

Standard Equation Forms

• Horizontal major axis (center $(h,k)$, $a > b$): $$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$

  • $a$ = semi-major axis length, $b$ = semi-minor axis length
  • Foci at $(h \pm c, k)$, where $c^2 = a^2 - b^2$

• Vertical major axis: $$\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$$

Example Analysis

• Given: $\frac{(x - 2)^2}{36} + \frac{(y + 2)^2}{4} = 1$
  • Center: $(2, -2)$
  • $a^2 = 36 \Rightarrow a = 6$, $b^2 = 4 \Rightarrow b = 2$
  • Major axis horizontal since $a > b$ under $x$ term
  • Vertices: $(2 \pm 6, -2)$ → $(8, -2)$ and $(-4, -2)$
  • Co-vertices: $(2, -2 \pm 2)$ → $(2, 0)$ and $(2, -4)$
  • $c = \sqrt{36 - 4} = \sqrt{32} = 4\sqrt{2}$
  • Foci: $(2 \pm 4\sqrt{2}, -2)$

Key Properties

• Eccentricity: $e = \frac{c}{a} \quad (0 < e < 1)$
• Axes: $2a$ (major axis length), $2b$ (minor axis length)
• Relationship: In any ellipse, $a^2 = b^2 + c^2$