Today in Precalculus• Go over homework• Notes: Hyperbolas

Completing the squareEccentricity

• Homework

HyperbolasProve that the graph of 9x2 – 4y2 – 18x + 8y + 41 = 0 is an hyperbola. Find the center, vertices and foci. Then graph the hyperbola by hand.

9x2 – 18x – 4y2 + 8y = -41

9(x2 – 2x) – 4(y2 – 2y) = -41

9(x2 – 2x + 1) – 4(y2 – 2y + 1) = -41 + 9 - 4

9(x – 1)2 – 4(y – 1)2 = -362 29( 1) 4( 1) 36

36 36 36

x y

2 2( 1) ( 1)

19 4

y x

x

y

Center: (1, 1)

a = ±3

Vertices: (1, -2), (1, 4)

c2 = 9 +4 = 13

c = ±3.6

Foci: (1, -2.6), (1,4.6 )

Pts on conjugate axis

b = ±2

(-1,1), (3,1)

2 2( 1) ( 1)1

9 4

y x

x

y

x

y

x

y

3: ( 1) 1

2Asymptotes y x

Example 2Prove that the graph of 9y2 – 25x2 + 72y – 100x +269 = 0 is an hyperbola. Find the center, vertices and foci. Then graph the hyperbola by hand.

9y2 + 72y – 25x2 – 100x = -269

9(y2 + 8y) – 25(x2 + 4x) = - 269

9(y2 + 8y + 16) – 25(x2 + 4x + 4) = -269 +144 -100

9(y + 4)2 – 25(x + 2)2 = -2252 29( 4) 25( 2) 225

225 225 225

y x

2 2( 2) ( 4)

19 25

x y

Center: (-2, -4)

a = ± 3

Vertices: (-5, -4), (1, -4)

c2 = 9 + 25 =34

c = ±5.8

Foci: (-7.8, -4), (3.8, -4)

Pts on conjugate axis

b = ±5

(-2, -9), (-2, 1)

2 2( 2) ( 4)1

9 25

x y

5: ( 2) 4

3Asymptotes y x

Eccentricity

Where a is the semitranvserse axis and

b is the semiconjugate axis.

Example a:

Example b:

2 2c a be

a a

9 49e 2.539

3

16 25e 1.601

4

Homework

Page 664: 47-50

Ellipse and Hyperbola Quiz: Tuesday, February 26