Module 7 Quick Reference
Conic Sections - One-Page Cheat Sheet
All Conic Sections - Quick Comparison
| Conic | Standard Form | Key Features |
|---|---|---|
| Circle | (x - h)2 + (y - k)2 = r2 | Center: (h, k), Radius: r |
| Parabola (V) | (x - h)2 = 4p(y - k) | Vertex: (h, k), Focus: (h, k+p) |
| Parabola (H) | (y - k)2 = 4p(x - h) | Vertex: (h, k), Focus: (h+p, k) |
| Ellipse (H) | (x-h)2/a2 + (y-k)2/b2 = 1, a > b | Center: (h, k), Vertices: (h±a, k) |
| Ellipse (V) | (x-h)2/b2 + (y-k)2/a2 = 1, a > b | Center: (h, k), Vertices: (h, k±a) |
| Hyperbola (H) | (x-h)2/a2 - (y-k)2/b2 = 1 | Center: (h, k), Vertices: (h±a, k) |
| Hyperbola (V) | (y-k)2/a2 - (x-h)2/b2 = 1 | Center: (h, k), Vertices: (h, k±a) |
Circles
Standard Form
(x - h)2 + (y - k)2 = r2
Center and Radius
Center: (h, k)
Radius: r
At Origin
x2 + y2 = r2
Center: (0, 0)
Completing the Square: For x2 + bx, add (b/2)2 to both sides
Parabolas
Vertical (Up/Down)
(x - h)2 = 4p(y - k)
Vertex: (h, k)
Focus: (h, k + p)
Directrix: y = k - p
Opens up if p > 0
Opens down if p < 0
Horizontal (Left/Right)
(y - k)2 = 4p(x - h)
Vertex: (h, k)
Focus: (h + p, k)
Directrix: x = h - p
Opens right if p > 0
Opens left if p < 0
At the Origin
- Vertical: x2 = 4py
- Horizontal: y2 = 4px
Ellipses
Horizontal Major Axis
(x-h)2/a2 + (y-k)2/b2 = 1
where a > b
Center: (h, k)
Vertices: (h ± a, k)
Co-vertices: (h, k ± b)
Foci: (h ± c, k)
Vertical Major Axis
(x-h)2/b2 + (y-k)2/a2 = 1
where a > b
Center: (h, k)
Vertices: (h, k ± a)
Co-vertices: (h ± b, k)
Foci: (h, k ± c)
Key Relationship
a2 = b2 + c2
Identify major axis: Larger denominator indicates major axis direction
Eccentricity
e = c/a where 0 < e < 1
Hyperbolas
Horizontal (Opens L/R)
(x-h)2/a2 - (y-k)2/b2 = 1
Center: (h, k)
Vertices: (h ± a, k)
Foci: (h ± c, k)
Asymptotes:
y - k = ±(b/a)(x - h)
Vertical (Opens U/D)
(y-k)2/a2 - (x-h)2/b2 = 1
Center: (h, k)
Vertices: (h, k ± a)
Foci: (h, k ± c)
Asymptotes:
y - k = ±(a/b)(x - h)
Key Relationship
c2 = a2 + b2
Note: For hyperbolas, c2 = a2 + b2 (different from ellipses!)
Asymptotes at the Origin
- Horizontal hyperbola: y = ±(b/a)x
- Vertical hyperbola: y = ±(a/b)x
How to Identify Conic Sections
Given: Ax2 + Cy2 + Dx + Ey + F = 0
| Conic | Condition |
|---|---|
| Circle | A = C (both same sign) |
| Parabola | A = 0 or C = 0 (one variable not squared) |
| Ellipse | A ≠ C, but both have same sign |
| Hyperbola | A and C have opposite signs |
Quick Check: Look at coefficients of x2 and y2 first!
Graphing Tips
Circles
- Plot center (h, k)
- Use radius r to find 4 points
- Draw smooth circle
Parabolas
- Plot vertex (h, k)
- Plot focus
- Determine opening direction
- Sketch U-shape
Ellipses
- Plot center (h, k)
- Plot vertices (use a)
- Plot co-vertices (use b)
- Draw smooth oval
Hyperbolas
- Plot center (h, k)
- Plot vertices (use a)
- Draw asymptotes
- Sketch branches approaching asymptotes
Common Formulas Summary
Distance and Midpoint
Distance: d = √[(x2 - x1)2 + (y2 - y1)2]
Midpoint: M = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint: M = ((x1 + x2)/2, (y1 + y2)/2)
Completing the Square
x2 + bx → (x + b/2)2 - (b/2)2
Relationship Formulas
| Conic | Relationship |
|---|---|
| Ellipse | a2 = b2 + c2 |
| Hyperbola | c2 = a2 + b2 |
Problem-Solving Steps
Converting to Standard Form
- Identify conic type from coefficients
- Group x terms and y terms
- Factor out coefficients if needed
- Complete the square for x and y
- Simplify to standard form
- Read off key features
Finding Key Features
- Identify center/vertex (h, k) from equation
- Find a, b, c, p, or r as appropriate
- Calculate vertices, foci, or directrix
- For hyperbolas, find asymptotes
- Verify orientation (horizontal vs. vertical)
Essential Tips
- Sign awareness: (x - h) means positive h, (x + h) means negative h
- Square roots: Always take square root of denominator to find a or b
- Ellipses: Larger denominator indicates major axis direction
- Hyperbolas: Positive term indicates opening direction
- Remember: Ellipse uses a2 = b2 + c2, Hyperbola uses c2 = a2 + b2
- Asymptotes: Draw asymptotes first for hyperbolas
- Completing the square: Add the same amount to both sides
- Check your work: Substitute a point to verify equation