Conic Sections Introduction

What are Conic Sections?

Conic sections: Curves formed by intersecting a plane with a double cone

Four types:

1. Circle: Plane perpendicular to cone's axis
2. Ellipse: Plane at angle, cuts one cone
3. Parabola: Plane parallel to side of cone
4. Hyperbola: Plane cuts both cones

All share common geometric properties

Circles

Definition: Set of all points equidistant from center

Standard form: (x - h)² + (y - k)² = r²

Where:

• (h, k) = center
• r = radius

Example 1: Write Equation

Center (3, -2), radius 5

Equation:

(x - 3)² + (y + 2)² = 25

Example 2: Find Center and Radius

Equation: (x + 1)² + (y - 4)² = 9

Rewrite:

(x - (-1))² + (y - 4)² = 3²

Center: (-1, 4) Radius: 3

Example 3: Circle at Origin

Center (0, 0), radius r

Simplified equation: x² + y² = r²

Example: x² + y² = 16 has center (0, 0) and radius 4

Parabolas

Definition: Set of points equidistant from focus and directrix

Two orientations:

Vertical (opens up/down):

• Standard form: (x - h)² = 4p(y - k)
• Vertex: (h, k)
• Focus: (h, k + p)
• Opens up if p > 0, down if p < 0

Horizontal (opens left/right):

• Standard form: (y - k)² = 4p(x - h)
• Vertex: (h, k)
• Focus: (h + p, k)
• Opens right if p > 0, left if p < 0

Example 1: Vertical Parabola

Vertex at origin, focus at (0, 2)

p = 2, so 4p = 8

Equation: x² = 8y

Opens upward

Example 2: From Vertex Form

y = (x - 2)² + 3

Rewrite in standard form:

(x - 2)² = y - 3

Vertex: (2, 3) Opens upward (coefficient of (x - 2)² is positive)

Example 3: Horizontal Parabola

Vertex (1, -2), focus (4, -2)

Horizontal, p = 3

Equation:

(y + 2)² = 12(x - 1)

Opens right

Ellipses

Definition: Set of points where sum of distances to two foci is constant

Standard form (horizontal major axis): (x - h)²/a² + (y - k)²/b² = 1 where a > b

Standard form (vertical major axis): (x - h)²/b² + (y - k)²/a² = 1 where a > b

Where:

• (h, k) = center
• a = semi-major axis (larger)
• b = semi-minor axis (smaller)
• c² = a² - b² (distance from center to focus)

Vertices: a units from center along major axis

Co-vertices: b units from center along minor axis

Foci: c units from center along major axis

Example 1: Horizontal Ellipse

Equation: x²/25 + y²/9 = 1

Center: (0, 0) a² = 25, so a = 5 (horizontal) b² = 9, so b = 3 (vertical)

Vertices: (±5, 0) Co-vertices: (0, ±3)

Find c:

c² = 25 - 9 = 16
c = 4

Foci: (±4, 0)

Example 2: Vertical Ellipse

Equation: x²/16 + y²/25 = 1

a² = 25 > b² = 16, so vertical

a = 5, b = 4

Center: (0, 0) Vertices: (0, ±5) Co-vertices: (±4, 0)

Foci:

c² = 25 - 16 = 9, c = 3
Foci: (0, ±3)

Example 3: Translated Ellipse

Center (2, -1), a = 6 (horizontal), b = 4

Equation:

(x - 2)²/36 + (y + 1)²/16 = 1

Hyperbolas

Definition: Set of points where absolute difference of distances to two foci is constant

Standard form (horizontal transverse axis): (x - h)²/a² - (y - k)²/b² = 1

Standard form (vertical transverse axis): (y - k)²/a² - (x - h)²/b² = 1

Where:

• (h, k) = center
• a = distance from center to vertex
• c² = a² + b² (distance to focus)
• Asymptotes pass through center with slopes ±b/a

Key difference from ellipse: Plus becomes minus, and c² = a² + b²

Example 1: Horizontal Hyperbola

Equation: x²/16 - y²/9 = 1

Center: (0, 0) a² = 16, a = 4 b² = 9, b = 3

Vertices: (±4, 0)

Foci:

c² = 16 + 9 = 25, c = 5
Foci: (±5, 0)

Asymptotes: y = ±(3/4)x

Example 2: Vertical Hyperbola

Equation: y²/25 - x²/16 = 1

Vertical (y² term positive)

a² = 25, a = 5 b² = 16, b = 4

Vertices: (0, ±5)

Foci:

c² = 25 + 16 = 41
c = √41
Foci: (0, ±√41)

Asymptotes: y = ±(5/4)x

Example 3: Asymptote Slopes

For hyperbola centered at origin:

Horizontal: y = ±(b/a)x

Vertical: y = ±(a/b)x

Summary of Standard Forms

Circle:

• (x - h)² + (y - k)² = r²

Parabola:

• Vertical: (x - h)² = 4p(y - k)
• Horizontal: (y - k)² = 4p(x - h)

Ellipse:

• (x - h)²/a² + (y - k)²/b² = 1 (a > b)

Hyperbola:

• (x - h)²/a² - (y - k)²/b² = 1 (horizontal)
• (y - k)²/a² - (x - h)²/b² = 1 (vertical)

Identifying Conic from Equation

General form: Ax² + Bxy + Cy² + Dx + Ey + F = 0

If B = 0 (no xy term):

Circle: A = C (coefficients equal)

Parabola: A = 0 or C = 0 (only one squared term)

Ellipse: A and C same sign, A ≠ C

Hyperbola: A and C opposite signs

Example: Identify Conic

4x² + 9y² = 36

Divide by 36:

x²/9 + y²/4 = 1

Plus sign, different denominators → Ellipse

x² = 8y

Only one squared term → Parabola

x² - y² = 1

Minus sign → Hyperbola

Eccentricity

Eccentricity (e): Measure of how "stretched" conic is

Circle: e = 0 Ellipse: 0 < e < 1 Parabola: e = 1 Hyperbola: e > 1

For ellipse: e = c/a

For hyperbola: e = c/a

Example: Find Eccentricity

Ellipse: x²/25 + y²/9 = 1

a = 5, b = 3 c² = 25 - 9 = 16, c = 4

Eccentricity:

e = 4/5 = 0.8

Fairly stretched ellipse

Real-World Applications

Astronomy: Planetary orbits (ellipses)

Engineering: Reflector shapes (parabolas, ellipses)

Architecture: Arches, domes

Physics: Particle paths, optics

Navigation: LORAN system (hyperbolas)

Example: Satellite Orbit

Earth at focus of elliptical orbit

Equation might be: x²/a² + y²/b² = 1

Where a > b, giving elongated orbit

Example: Parabolic Reflector

Satellite dish, headlight reflector

Parabola reflects parallel rays to focus

Design equation: x² = 4py

Completing the Square

Convert general form to standard form

Example: Complete the Square

x² + y² - 4x + 6y - 3 = 0

Group terms:

(x² - 4x) + (y² + 6y) = 3

Complete squares:

(x² - 4x + 4) + (y² + 6y + 9) = 3 + 4 + 9
(x - 2)² + (y + 3)² = 16

Circle: center (2, -3), radius 4

Practice