Conic Sections Formulas

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Algebra

28 Terms

1

Circle

(x-h)²+(y-k)²=r²

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2

Area of a Circle

πr²

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3

Degenerate Circle

(x-h)²+(y-k)²=0

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4

How can you tell if an equation is a circle without completing the square?

  • both variables are squared, somewhere in the equation

  • both x² and y² have exactly the same coefficient and the same sign

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5

Parabola (up/down)

y= a(x - h)² +k

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6

Parabola (left/right)

x= a(y - k)² +h

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7

Parabola (a)

a = 1/4c

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8

Parabola (c)

c = 1/4a

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9

Length of Latus Rectum (parabola)

I 1/a I = I 4c I

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10

C= dist. from . . .

vertex to focus or from vertex to directrix

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11

How can you tell if an equation is a parabola without completing the square?

  • only x or y is squared, but not both

  • if x is squared, then the graph goes up (a>0) or down (a<0)

  • if y is squared, then the graph goes right (a>0) or left (a<0)

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12

Ellipses formula (stretched horizontally)

(x - h)² + (y - k)² = 1

a² b²

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13

Ellipses formula (stretched vertically)

(y - k)² + (x - h)² = 1

a² b²

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14

Area of interior of ellipse

πab

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15

Sum of focal radii

2a

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16

length of major axis (ellipses)

2a

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17

length of minor axis

2b

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18

C= distance from . . .

center to each of the foci

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19

a²= (ellipses)

b² + c²

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20

How do you know if an equation is an ellipse without completing the square?

  • both of the variables are squared, somewhere in the equation

  • coefficients of x² and y² are different

  • signs of the coefficients of x² and y² are the same

  • the square root of the number under the “x” term will indicate the horizontal stretch

  • the square root of the number under the “y” term will indicate the vertical stretch

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21

Hyperbola formula (left/right)

[(x - h)²/ a²] - [(y - k)²/ b²] = 1

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22

hyperbola formula (up/down)

[(y - k)²/ a²] - [(x - h)²/ b²] = 1

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23

Difference of focal radii

2a

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24

c= distance from . . . . (hyperbola)

center to each focus

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25

c²=

a² + b²

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26

How do you know if an equation is a hyperbola without completing the square?

  • both variables are squared, somewhere in the equation

  • coefficients of x² and y² may be the same or may be different

  • signs of coefficients of x² and y² are different

  • if the x term is positive/first, then the graph is “stretched” horizontally, if the y term is positive/first then the graph is “stretched” vertically

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27

Degenerate Hyperbola

[(x - h)²/ a²] - [(y - k)²/ b²] = 0

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28

Degenerate Ellipse

[(x - h)²/ a²] + [(y - k)²/ b²] = 0

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