11.3 The Hyperbola

Hyperbola: the set of all points P in a plane such that the absolute value of the difference of the distances from two fixed points is a constant.

(foci)

2 2

2 2 1x ya b

Standard Equation (centered at origin)

x-int: (±a, 0)y-int: nonetransverse axis: 2aconjugate axis: 2bfoci: (±c, 0) c2 = a2 + b2

asymptotes:

2 2

2 2 1y xa b

by xa

x-int: noney-int: (0, ±a)transverse axis: 2aconjugate axis: 2bfoci: (0, ±c) c2 = a2 + b2

asymptotes: ay xb

eqtns of asym.:

y in front vertices (0, ±3)endpoints of conjugate axis: (±4, 0)Foci: c2 = 9 + 16 = 25 c = 5 (on y-axis) (0, ±5)

distance from center to a focusdistance from center to a vertex

cea

Eccentricity is still ratio:

e > 1 iff it is a hyperbola

Ex 1) Determine the vertices, endpoints of conjugate axis, the foci, the

asymptotes and eccentricity of . Graph it.*Hint: Graph as you go!

2 2

19 16y x

34

y x

53

cea

What if center is not the origin? Standard form of a hyperbola with center (h, k) & with axes parallel to

the coordinate axes is2 2 2 2

2 2 2 2

( ) ( ) ( ) ( )1 1x h y k y k x ha b a b

OR

Ex 2) Graph . Determine center, vertices, foci, & asymptotes.

Center (4, –3)Eqtns of asym – use point slope with center

&

x in front – vertices: (4 ± 5, –3) (9, –3) (–1, –3)

endpoints of conjugate axis: (4, –3 ± 4) (4, 1) (4, –7)

2 2( 4) ( 3) 125 16

x y

2 25 16 41 41 4 41, 3c c

453 ( 4)bm y x

a

Ex 3) Determine the center, vertices, foci and asymptotes of 3x2 – y2 – 12x – 6y = 0. Then graph it.

3(x2 – 4x + 4 ) – (y2 + 6y + 9 ) = 0 + 12 + –9

3(x – 2)2 – (y + 3)2 = 32 2( 2) ( 3) 1

1 3x y

Center (2, –3)Eqtns of asym: x in front – vertices: (2 ± 1, –3)

(3, –3) (1, –3)endpts of conj axis:

Foci: c2 = 1 + 3 = 4 c = 2 (2 ± 2, –3) (4, –3) (0, –3)

2, 3 3 2, 1.3 2, 4.7

3 3 2y x

2 2

2

2 2

2

2

2 2

2

2

2

( 3) 125

( 16 3) (12) 125

169 144 125

169 3600 25

3

25

600 144

25

y xb

b

b

b b

b

b

b

Ex 4) Determine an equation in standard form for a hyperbola with vertices at (0, 2) and (0, –8) with points (12, –16) and being two points on the hyperbola.

center halfway between vertices½ (2 – 8) = ½(–6) = –3C(0, –3) a = 5 on y-axis

6, 61 3

(12, –16):

plug in one of the points to solve for b2

2 2( 3) 125 25

y x

Homework

#1103 Pg 555 #1, 5, 9, 14, 17, 21, 29, 33, 36, 37