AP Precalculus: Conic Sections Formulas & Properties
1. Parabola
- Vertex form (vertical): \( y = a(x - h)^2 + k \)
- Vertex: \( (h, k) \)
- Focus: \( (h, k + \frac{1}{4a}) \)
- Directrix: \( y = k - \frac{1}{4a} \)
- Axis of symmetry: \( x = h \)
- Horizontal parabolas: \( x = a(y-k)^2 + h \)
2. Circle
- Standard form: \( (x-h)^2 + (y-k)^2 = r^2 \)
- Center: \( (h, k) \)
- Radius: \( r \)
- All circles have eccentricity \( e = 0 \)
3. Ellipse
- Standard (horizontal): \( \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \), \( a > b \)
- Center: \( (h, k) \)
- Vertices: \( (h\pm a, k) \)
- Foci: \( (h\pm c, k) \) where \( c = \sqrt{a^2-b^2} \)
- Minor axis: length \( 2b \)
- Major axis: length \( 2a \)
- Eccentricity: \( e = \frac{c}{a} \), \( 0 < e < 1 \)
- Vertical ellipse: switch \( a \) and \( b \) positions (major axis along y)
4. Hyperbola
- Standard (horizontal): \( \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \)
- Center: \( (h, k) \)
- Vertices: \( (h\pm a, k) \)
- Foci: \( (h\pm c, k) \), \( c = \sqrt{a^2+b^2} \)
- Transverse axis: length \( 2a \)
- Conjugate axis: length \( 2b \)
- Eccentricity: \( e = \frac{c}{a} \), \( e > 1 \)
- Vertical hyperbola: \( \frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1 \)
- Asymptotes: \( y-k = \pm \frac{b}{a}(x-h) \)
5. Convert to Standard Form
- General conic: \( Ax^2 + By^2 + Cx + Dy + E = 0 \)
- Complete the square for \( x \) and \( y \)
- Divide both sides and arrange as required to match standard form for each conic