The document describes the key properties of a hyperbola with eccentricity e > 1. It defines the hyperbola equation and explains that the foci lie on the y-axis. It also derives formulas for the eccentricity, semi-major and semi-minor axes, foci, directrices, and asymptotes. An example is provided to demonstrate calculating these values for a specific hyperbola.

1. Hyperbola (e >1)

2. Hyperbola (e >1)
y
-a a
Hyperbola: x2 y2
 2 1
where; b 2  a 2 e 2  1 2
a b

3. Hyperbola (e >1)
y
S’(-ae,0) -a a S(ae,0)
Hyperbola: x2 y2
 2 1
where; b 2  a 2 e 2  1 2
a b
focus :  ae,0 

4. Hyperbola (e >1)
y
S’(-ae,0) -a a S(ae,0)
a a
x x
e e
Hyperbola: x2 y2
 2 1
where; b 2  a 2 e 2  1 2
a b
focus :  ae,0 
a
directrices : x  
e

5. Hyperbola (e >1) b
b y y x
y x
a a
S’(-ae,0) -a a S(ae,0) x
a a
x x
e e
Hyperbola: x2 y2
 2 1
where; b 2  a 2 e 2  1 2
a b
focus :  ae,0 
a b
directrices : x   asymptotes : y   x
e a

6. Hyperbola (e >1) b
b y y x
y x
a a
S’(-ae,0) -a a S(ae,0) x
y2 x2
Note : If 2  2  1
a a b a
x x foci on the y axis
e e
Hyperbola: x2 y2
 2 1
where; b 2  a 2 e 2  1 2
a b
focus :  ae,0 
a b
directrices : x   asymptotes : y   x
e a

7. Hyperbola (e >1) b
b y y x
y x
a a
S’(-ae,0) -a a S(ae,0) x
y2 x2
Note : If 2  2  1
a a b a
x x foci on the y axis
e e
Hyperbola: a 2  b 2 e 2  1
x2 y2
 2 1 focus : 0,be 
where; b 2  a 2 e 2  1 2
a b
focus :  ae,0  directrices : y  
b
a b e
directrices : x   asymptotes : y   x b
e a asymptotes : y   x
a

8. e.g. Find the eccentricity, foci, directrices and asymptotes of the
x2 y2
hyperbola   1
9 16

9. e.g. Find the eccentricity, foci, directrices and asymptotes of the
x2 y2
hyperbola   1
9 16
a2  9
a3

10. e.g. Find the eccentricity, foci, directrices and asymptotes of the
x2 y2
hyperbola   1
9 16
a 9
2 b 2  16
a3 a 2 e 2  1  16

11. e.g. Find the eccentricity, foci, directrices and asymptotes of the
x2 y2
hyperbola   1
9 16
a 9
2 b 2  16
a3 a 2 e 2  1  16
9e 2  1  16
16
e 1 
2
9
25
e 
2
9
5
e
3

12. e.g. Find the eccentricity, foci, directrices and asymptotes of the
x2 y2
hyperbola   1
9 16
5
b 2  16  eccentricity 
a 9
2
3
a3 a 2 e 2  1  16 foci :  5,0 
9e 2  1  16 directrices : x  
9
16 5
e 1 
2
4
9 asymptotes : y   x
3
25
e 
2
9
5
e
3

13. e.g. Find the eccentricity, foci, directrices and asymptotes of the
x2 y2
hyperbola   1
9 16
5
b 2  16  eccentricity 
a 9
2
3
a3 a 2 e 2  1  16 foci :  5,0 
9e 2  1  16 directrices : x  
9
16 5
e 1 
2
4
9 asymptotes : y   x
3
25
e 
2
9
5
e Exercise 6B; 1acd, 2, 3
3