Class Notes: Conic Sections - Key Formulas and Concepts

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1. Circle

Standard Equation

• Standard form: [(x - h)^2 + (y - k)^2 = r^2]
  • Center: ((h, k))
  • Radius: (r)

Key Properties

• Eccentricity: 0
• Area: (\pi r^2)
• Circumference: (2\pi r)
• Diameter: (2r)

Domain and Range

• Domain: (h - r \leq x \leq h + r)
• Range: (k - r \leq y \leq k + r)

Semicircle Equations

• Solve for (y): (y = k \pm \sqrt{r^2 - (x - h)^2})
  • (+) is the upper half, (-) is the lower half.
• Solve for (x): (x = h \pm \sqrt{r^2 - (y - k)^2})
  • (+) is the right half, (-) is the left half.

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2. Ellipse

Standard Equations

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

Dimensions

• Major axis length: (2a)
• Minor axis length: (2b)
• Center: ((h, k))

Foci & Eccentricity

• Pythagorean relationship: (c^2 = a^2 - b^2)
• Eccentricity: (e = \frac{c}{a}), (0 < e < 1)
• Foci:
  • Horizontal ellipse: ((h \pm c, k))
  • Vertical ellipse: ((h, k \pm c))
• Vertices:
  • Major: ((h \pm a, k)) or ((h, k \pm a))
  • Minor: ((h, k \pm b)) or ((h \pm b, k))

Area & Approximate Circumference

• Area: (\pi ab)
• Circumference (approx.): (\pi(a + b)) or more accurately: [2\pi \sqrt{\frac{a^2 + b^2}{2}}]

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3. Hyperbola

Standard Equations

• Horizontal transverse axis: [\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1]
• Vertical transverse axis: [\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1]

Properties

• Center: ((h, k))
• Vertices: ((h \pm a, k)) (horizontal) or ((h, k \pm a)) (vertical)
• Transverse axis (real axis): along the (a) term (positive)
• Pythagorean relationship: (c^2 = a^2 + b^2)
• Eccentricity: (e = \frac{c}{a}), (e > 1)
• Foci:
  • Horizontal: ((h \pm c, k))
  • Vertical: ((h, k \pm c))

Asymptotes

• Horizontal: (y - k = \pm \frac{b}{a}(x - h))
• Vertical: (y - k = \pm \frac{a}{b}(x - h))

Domain and Range

• For horizontal: domain excludes values between the vertices; range is all real numbers.
• For vertical: range excludes values between the vertices; domain is all real numbers.

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4. Parabola

Standard Equations

• Opens right/left: ((y - k)^2 = 4p(x - h))
• Opens up/down: ((x - h)^2 = 4p(y - k))
• Vertex: ((h, k))
• Eccentricity: 1

Direction and Focus

• Opens:
  • Up/right if (p > 0)
  • Down/left if (p < 0)
• Focus:
  • For ((y - k)^2): ((h + p, k))
  • For ((x - h)^2): ((h, k + p))
  • Distance from vertex to focus: (|p|)

Directrix

• For ((y - k)^2): (x = h - p)
• For ((x - h)^2): (y = k - p)

Axis of Symmetry

• Horizontal: (y = k)
• Vertical: (x = h)

Latus Rectum

• Length: (4p)
• Passes through focus, parallel to directrix

Intercepts

• Right/left opening: Single x-intercept (= \frac{k^2}{4p} + h)
• Up/down opening: Potentially two x-intercepts: (x = h \pm \sqrt{-4pk} )
• Similar formula applies for y-intercepts

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Additional Notes

• Formula Sheets: More formulas for domains, ranges, intercepts, and graphs available as printable resources (see video description or provided materials).
• Reference Videos: Links to additional video tutorials for graphing and examples are available.

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End of Notes