Study Guide - Module 7 - College Algebra

1. Introduction to Conic Sections

Conic sections are curves formed by the intersection of a plane and a double cone. The four types are: circles, parabolas, ellipses, and hyperbolas.

1.1 General Equation

Ax² + Bxy + Cy² + Dx + Ey + F = 0

1.2 How to Identify Conic Sections

Circle: Coefficients of x² and y² are equal (A = C) and same sign
Parabola: Only one variable is squared (A = 0 or C = 0, but not both)
Ellipse: Coefficients of x² and y² have the same sign but are not equal (A ≠ C)
Hyperbola: Coefficients of x² and y² have opposite signs

2. Circles

2.1 Definition

A circle is the set of all points in a plane that are equidistant from a fixed point (the center).

2.2 Standard Form

(x - h)² + (y - k)² = r²

Where:

(h, k) = center of the circle
r = radius of the circle

Example: (x - 3)² + (y + 5)² = 16
Center: (3, -5)
Radius: r = 4 (since r² = 16)

2.3 Completing the Square

To convert from general form to standard form:

Group x terms and y terms separately
Complete the square for x: add (b/2)² where b is the coefficient of x
Complete the square for y: add (d/2)² where d is the coefficient of y
Add the same amounts to the right side of the equation
Factor to get standard form

Example: x² + y² - 6x + 4y - 3 = 0
(x² - 6x) + (y² + 4y) = 3
(x² - 6x + 9) + (y² + 4y + 4) = 3 + 9 + 4
(x - 3)² + (y + 2)² = 16
Center: (3, -2), Radius: 4

2.4 Key Formulas

Distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

3. Parabolas

3.1 Definition

A parabola is the set of all points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix).

3.2 Vertical Parabolas (Opens Up or Down)

(x - h)² = 4p(y - k)

Where:

(h, k) = vertex
p = distance from vertex to focus (and from vertex to directrix)
If p > 0, opens upward; if p < 0, opens downward
Focus: (h, k + p)
Directrix: y = k - p

Example: (x - 2)² = 8(y + 1)
Vertex: (2, -1)
4p = 8, so p = 2
Focus: (2, -1 + 2) = (2, 1)
Directrix: y = -1 - 2 = -3

3.3 Horizontal Parabolas (Opens Left or Right)

(y - k)² = 4p(x - h)

Where:

(h, k) = vertex
p = distance from vertex to focus
If p > 0, opens right; if p < 0, opens left
Focus: (h + p, k)
Directrix: x = h - p

Example: (y - 3)² = -12(x + 2)
Vertex: (-2, 3)
4p = -12, so p = -3
Focus: (-2 + (-3), 3) = (-5, 3)
Directrix: x = -2 - (-3) = 1

3.4 Special Form at the Origin

Vertical: x² = 4py
Horizontal: y² = 4px

4. Ellipses

4.1 Definition

An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (foci) is constant.

4.2 Horizontal Ellipse (Major Axis Horizontal)

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

Where:

(h, k) = center
a = distance from center to vertex (on major axis)
b = distance from center to co-vertex (on minor axis)
c = distance from center to focus
Relationship: a² = b² + c²
Vertices: (h ± a, k)
Co-vertices: (h, k ± b)
Foci: (h ± c, k)

Example: (x - 1)²/25 + (y + 2)²/9 = 1
Center: (1, -2)
a = 5, b = 3
c² = 25 - 9 = 16, so c = 4
Vertices: (-4, -2) and (6, -2)
Foci: (-3, -2) and (5, -2)

4.3 Vertical Ellipse (Major Axis Vertical)

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

Where:

Vertices: (h, k ± a)
Co-vertices: (h ± b, k)
Foci: (h, k ± c)

4.4 Eccentricity

e = c/a where 0 < e < 1

Eccentricity measures how "stretched" the ellipse is. Closer to 0 = more circular, closer to 1 = more elongated.

4.5 Special Form at the Origin

Horizontal: x²/a² + y²/b² = 1
Vertical: x²/b² + y²/a² = 1

5. Hyperbolas

5.1 Definition

A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points (foci) is constant.

5.2 Horizontal Hyperbola (Opens Left and Right)

(x - h)²/a² - (y - k)²/b² = 1

Where:

(h, k) = center
a = distance from center to vertex
b = used to find asymptotes
c = distance from center to focus
Relationship: c² = a² + b²
Vertices: (h ± a, k)
Foci: (h ± c, k)
Asymptotes: y - k = ±(b/a)(x - h)

Example: (x - 2)²/16 - (y + 1)²/9 = 1
Center: (2, -1)
a = 4, b = 3
c² = 16 + 9 = 25, so c = 5
Vertices: (-2, -1) and (6, -1)
Foci: (-3, -1) and (7, -1)
Asymptotes: y + 1 = ±(3/4)(x - 2)

5.3 Vertical Hyperbola (Opens Up and Down)

(y - k)²/a² - (x - h)²/b² = 1

Where:

Vertices: (h, k ± a)
Foci: (h, k ± c)
Asymptotes: y - k = ±(a/b)(x - h)

5.4 Special Form at the Origin

Horizontal: x²/a² - y²/b² = 1, Asymptotes: y = ±(b/a)x
Vertical: y²/a² - x²/b² = 1, Asymptotes: y = ±(a/b)x

5.5 Graphing Hyperbolas

Plot the center
Plot the vertices using a
Draw a rectangle using a and b from the center
Draw asymptotes through opposite corners of the rectangle
Sketch the hyperbola approaching the asymptotes

6. Summary of All Conic Sections

Conic	Standard Form (Centered at (h, k))	Key Features
Circle	(x - h)² + (y - k)² = r²	Center: (h, k) Radius: r
Vertical Parabola	(x - h)² = 4p(y - k)	Vertex: (h, k) Focus: (h, k + p) Directrix: y = k - p
Horizontal Parabola	(y - k)² = 4p(x - h)	Vertex: (h, k) Focus: (h + p, k) Directrix: x = h - p
Horizontal Ellipse	(x - h)²/a² + (y - k)²/b² = 1 (a > b)	Center: (h, k) Vertices: (h ± a, k) Foci: (h ± c, k) a² = b² + c²
Vertical Ellipse	(x - h)²/b² + (y - k)²/a² = 1 (a > b)	Center: (h, k) Vertices: (h, k ± a) Foci: (h, k ± c) a² = b² + c²
Horizontal Hyperbola	(x - h)²/a² - (y - k)²/b² = 1	Center: (h, k) Vertices: (h ± a, k) Foci: (h ± c, k) c² = a² + b² Asymptotes: y - k = ±(b/a)(x - h)
Vertical Hyperbola	(y - k)²/a² - (x - h)²/b² = 1	Center: (h, k) Vertices: (h, k ± a) Foci: (h, k ± c) c² = a² + b² Asymptotes: y - k = ±(a/b)(x - h)

7. How to Identify Conic Sections from Equations

7.1 Quick Identification Method

Given the general form: Ax² + Cy² + Dx + Ey + F = 0

Circle: A = C (coefficients equal and same sign)
Parabola: A = 0 or C = 0 (only one variable squared)
Ellipse: A ≠ C, but both same sign (both positive or both negative)
Hyperbola: A and C have opposite signs

Examples:
x² + y² - 4x + 6y = 12 → Circle (coefficients both 1)
y² - 8x + 4y = 16 → Parabola (only y is squared)
4x² + 9y² = 36 → Ellipse (4 ≠ 9, both positive)
9x² - 4y² = 36 → Hyperbola (opposite signs)

8. Problem-Solving Strategies

8.1 Converting to Standard Form

Identify the type of conic section
Group x terms and y terms together
Factor out coefficients if necessary
Complete the square for x and/or y
Simplify to match standard form
Identify key features (center, radius, vertices, foci, etc.)

8.2 Graphing Conics

Convert to standard form if needed
Identify and plot the center (or vertex for parabolas)
Plot key points (vertices, co-vertices, or focus)
For hyperbolas, draw asymptotes first
Sketch the curve smoothly through the points

8.3 Common Mistakes to Avoid

Watch out for:
• Sign errors when finding h and k (remember x - h and y - k)
• Confusing a² with a (remember to take the square root)
• For ellipses: mixing up a² = b² + c² with hyperbolas c² = a² + b²
• Forgetting to check which denominator is larger for ellipses
• For hyperbolas, confusing the asymptote formulas for horizontal vs. vertical

9. Applications of Conic Sections

9.1 Real-World Uses

Circles: Wheels, gears, circular motion, ripples in water
Parabolas: Satellite dishes, headlight reflectors, projectile motion, suspension bridges
Ellipses: Planetary orbits, whispering galleries, medical lithotripsy
Hyperbolas: Navigation systems (LORAN), cooling towers, sonic booms

9.2 Reflective Property of Parabolas

All rays parallel to the axis of symmetry reflect off a parabolic mirror and pass through the focus. This is why satellite dishes and headlights use parabolic shapes.

9.3 Foci Property of Ellipses

A ray from one focus of an ellipse reflects off the ellipse and passes through the other focus. This creates "whispering galleries" where sound from one focus can be heard clearly at the other focus.

10. Test-Taking Tips

Identify the conic type first before attempting to solve
Check the signs of coefficients and constant terms carefully
For completing the square, remember to add to both sides
Write down known values (h, k, a, b, c, p, r) as you identify them
For ellipses, determine which axis is major (larger denominator)
For hyperbolas, note that c² = a² + b² (different from ellipses)
Draw a quick sketch to visualize the problem
Verify your answer makes sense in context
Remember the orientation: which variable is squared affects direction
For asymptotes, use the correct formula based on orientation

Module 7 Study Guide

1. Introduction to Conic Sections

1.1 General Equation

1.2 How to Identify Conic Sections

2. Circles

2.1 Definition

2.2 Standard Form

2.3 Completing the Square

2.4 Key Formulas

3. Parabolas

3.1 Definition

3.2 Vertical Parabolas (Opens Up or Down)

3.3 Horizontal Parabolas (Opens Left or Right)

3.4 Special Form at the Origin

4. Ellipses

4.1 Definition

4.2 Horizontal Ellipse (Major Axis Horizontal)

4.3 Vertical Ellipse (Major Axis Vertical)

4.4 Eccentricity

4.5 Special Form at the Origin

5. Hyperbolas

5.1 Definition

5.2 Horizontal Hyperbola (Opens Left and Right)

5.3 Vertical Hyperbola (Opens Up and Down)

5.4 Special Form at the Origin

5.5 Graphing Hyperbolas

6. Summary of All Conic Sections

7. How to Identify Conic Sections from Equations

7.1 Quick Identification Method

8. Problem-Solving Strategies

8.1 Converting to Standard Form

8.2 Graphing Conics

8.3 Common Mistakes to Avoid

9. Applications of Conic Sections

9.1 Real-World Uses

9.2 Reflective Property of Parabolas

9.3 Foci Property of Ellipses

10. Test-Taking Tips