Circles + Ellipses + Hyperbolas

Circles

Standard Formula → (x - h)² + (y - k)² = r²

**leave the radius as squared. DON’T square or divide it**

• Center → (opp. h, opp k)

• R → radius

  • if given such as → radius: 5

    • square it - becomes 25

General Formula → ex: x² + y² +4x - 6y + 12 = 0

1. Add constant

2. Rewrite so that x’s are grouped together and y’s are grouped together

3. complete the square w/ x’s

4. Factored to standard form

   • ***Can only complete the square if LC = 1***

Key Components

• Solution Point

  • Plug in to the x and y

    • Which would get you the r²

• Diameter Pts

  • First - find midpoint

    • (x₁ + x₂)/2, (y₁ + y₂)/2 = (x, y)

  • Choose one pt and plug in

    • would get you r²

• x-intercept

  • y = 0

    • (plug in 0 for y)

• y-intercept

  • x = 0

    • (plug in 0 for x)

Ellipses

Standard Form → (x - h)² / a² or b² + (y - k)² / a² or b² = 1

Order to solve:

1. Center

   1. ex: (-2, 2)

2. Vertices / Major axis

   1. Find a² and a

      1. after solved, + or - a with the greater value

         1. ex: Greater value = y THUS pts would be → (-2, 8) (-2, -4)

3. Foci

   1. a² - b² = c²

      1. Plug greater value to the center

4. Co-vertices / Minor axis

   1. Find b² and b

      1. + or - b with least value

Key Components

• must always set = 1

• a²

  • is the larger denominator

    • a = distance from center to vertex (2 vertices)

• b²

  • is the smaller denominator

    • distance from center to co-vertex (2 co-vertices)

• Foci

  • c = distance from center to focus pt (2 foci pts)

    • same direction as vertices

• Major Axis

  • 2a (is w/ a²)

    • Distance between vertices

• Minor Axis

  • 2b (is w/ b²)

    • distance between co-vertices

Hyperbola

Standard Form → (x - h)² / a² or b² - (y - k)² / a² or b² = 1

• Separated by subtraction

• opens in the direction of the positive term

Order to solve:

1. Center

2. Vertics

   1. a² and a

      1. plug in

3. B-Value

   1. b² and b

      1. NO PTS

4. Asymptotes

   1. pt: center

   2. slope ± y/x

      1. If slope doesn’t = the right # just keep on increasing the amount by 2

5. Foci

   1. a² + b² = c²

Key Components

• a²

  • under the POSITIVE TERM

• b² (= asymptotes)

  • under the NEGATIVE TERM

• Foci

  • opp. sign from equation

• Asymptotes (= b²)

  • y - k = ±y/x (x - h)

    • ryse/run

      • Center to the corner of the box

• Passes through the point: (5, ,4)

  • sub in and solve for missing piece

Identify the conic as circle, ellipse, or hyperbola

Circle

• Coef of x² and y² are both positive and same #

Ellipse

• Coef of x² and y² are both positive but NOT the same #

Hyperbola

• Coef of x² is negative