Conic Sections
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Parabolas
Standard Forms:
Vertical Parabola (opens up/down):
\[ (x - h)^2 = 4p(y - k) \]
• Vertex: (h, k)
• Focus: (h, k + p)
• Directrix: \( y = k - p \)
• Axis of symmetry: \( x = h \)
• Opens up if p > 0, down if p < 0
Horizontal Parabola (opens left/right):
\[ (y - k)^2 = 4p(x - h) \]
• Vertex: (h, k)
• Focus: (h + p, k)
• Directrix: \( x = h - p \)
• Axis of symmetry: \( y = k \)
• Opens right if p > 0, left if p < 0
Vertex Form:
\[ y = a(x - h)^2 + k \]
Vertex at (h, k), opens up if a > 0, down if a < 0
2. Circles
Standard Form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Properties:
• Center: (h, k)
• Radius: r
• Diameter: 2r
• Eccentricity: e = 0
Example:
Find center and radius: \( (x + 3)^2 + (y - 2)^2 = 25 \)
Center: (-3, 2)
Radius: \( r = \sqrt{25} = 5 \)
3. Ellipses
Standard Forms:
Horizontal Major Axis (wider):
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \quad (a > b) \]
• Center: (h, k)
• Vertices: (h ± a, k)
• Co-vertices: (h, k ± b)
• Foci: (h ± c, k) where \( c^2 = a^2 - b^2 \)
• Major axis length: 2a, Minor axis length: 2b
Vertical Major Axis (taller):
\[ \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \quad (a > b) \]
• Center: (h, k)
• Vertices: (h, k ± a)
• Co-vertices: (h ± b, k)
• Foci: (h, k ± c) where \( c^2 = a^2 - b^2 \)
Eccentricity:
\[ e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}} \]
For ellipse: 0 < e < 1 (circle has e = 0)
4. Hyperbolas
Standard Forms:
Horizontal Transverse Axis:
\[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \]
• Center: (h, k)
• Vertices: (h ± a, k)
• Foci: (h ± c, k) where \( c^2 = a^2 + b^2 \)
• Asymptotes: \( y - k = \pm\frac{b}{a}(x - h) \)
Vertical Transverse Axis:
\[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \]
• Center: (h, k)
• Vertices: (h, k ± a)
• Foci: (h, k ± c) where \( c^2 = a^2 + b^2 \)
• Asymptotes: \( y - k = \pm\frac{a}{b}(x - h) \)
Eccentricity:
\[ e = \frac{c}{a} = \sqrt{1 + \frac{b^2}{a^2}} \]
For hyperbola: e > 1
5. Conic Sections Summary
| Conic | Eccentricity | Standard Form |
|---|---|---|
| Circle | e = 0 | \( (x-h)^2 + (y-k)^2 = r^2 \) |
| Parabola | e = 1 | \( (x-h)^2 = 4p(y-k) \) |
| Ellipse | 0 < e < 1 | \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) |
| Hyperbola | e > 1 | \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) |
6. Important Relationships
For Ellipse: \( c^2 = a^2 - b^2 \) (c < a)
For Hyperbola: \( c^2 = a^2 + b^2 \) (c > a)
Where:
• a = semi-major axis (or distance to vertex)
• b = semi-minor axis
• c = distance from center to focus
7. Convert General to Standard Form
General Form:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
Steps to Convert:
1. Group x terms together and y terms together
2. Move constant to the right side
3. Complete the square for x terms
4. Complete the square for y terms
5. Factor perfect squares
6. Divide to get standard form (if needed)
Example:
Convert to standard form: \( x^2 + y^2 - 6x + 4y - 3 = 0 \)
Step 1: Group terms: \( (x^2 - 6x) + (y^2 + 4y) = 3 \)
Step 2: Complete the square for x: \( \left(\frac{-6}{2}\right)^2 = 9 \)
Step 3: Complete the square for y: \( \left(\frac{4}{2}\right)^2 = 4 \)
Step 4: \( (x^2 - 6x + 9) + (y^2 + 4y + 4) = 3 + 9 + 4 \)
Step 5: Factor: \( (x - 3)^2 + (y + 2)^2 = 16 \)
This is a circle with center (3, -2) and radius 4
8. Identifying Conic Types from Equations
From \( Ax^2 + Cy^2 + Dx + Ey + F = 0 \) (no xy term):
• Circle: A = C (same coefficients, same sign)
• Ellipse: A ≠ C, but both same sign (both + or both −)
• Hyperbola: A and C have opposite signs (one +, one −)
• Parabola: Either A = 0 or C = 0 (but not both)
9. Quick Reference Formulas
Key Formulas:
Circle: \( (x-h)^2 + (y-k)^2 = r^2 \)
Parabola: \( (x-h)^2 = 4p(y-k) \) or \( (y-k)^2 = 4p(x-h) \)
Ellipse: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
Hyperbola: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \)
Ellipse c-value: \( c^2 = a^2 - b^2 \)
Hyperbola c-value: \( c^2 = a^2 + b^2 \)
Eccentricity: \( e = \frac{c}{a} \)
📚 Study Tips
✓ For ellipse: c² = a² - b² (subtract); For hyperbola: c² = a² + b² (add)
✓ In standard form, larger denominator goes with major axis
✓ Parabola: distance from vertex to focus = |p|
✓ Eccentricity tells shape: 0=circle, <1=ellipse, 1=parabola, >1=hyperbola
✓ Complete the square to convert from general to standard form