2. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
3. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
4. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
e=0 circle
5. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
e=0 circle
e<1 ellipse
6. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
e=0 circle
e<1 ellipse
e=1 parabola
7. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
e=0 circle
e<1 ellipse
e=1 parabola
e>1 hyperbola
10. Ellipse (e < 1)
y
b
A’ A
-a S a Z x
-b
SA = eAZ and SA’ = eA’Z
11. Ellipse (e < 1)
y
b
A’ A
-a S a Z x
-b
SA = eAZ and SA’ = eA’Z
(1) SA’ + SA = 2a
(2) SA’ – SA = e(A’Z – AZ)
12. Ellipse (e < 1)
y
b
A’ A
-a S a Z x
-b
SA = eAZ and SA’ = eA’Z
(1) SA’ + SA = 2a
(2) SA’ – SA = e(A’Z – AZ)
= e(AA’)
= e(2a)
= 2ae
13. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e)
SA’ = a(1 + e)
14. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
15. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
Focus
OS = OA - SA
16. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
Focus
OS = OA - SA
= a – a(1 – e)
= ae
S ae,0
17. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
Focus Directrix
OS = OA - SA OZ = OA + AZ
= a – a(1 – e)
= ae
S ae,0
18. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
Focus Directrix
OS = OA - SA OZ = OA + AZ
SA
= a – a(1 – e) OA SA eAZ
= ae e
S ae,0
19. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
Focus Directrix
OS = OA - SA OZ = OA + AZ
SA
= a – a(1 – e) OA SA eAZ
= ae e
ae a1 e
S ae,0
e e a
a directrices x
e
e
20. S ae,0
b P
P x, y
N
A’ A
-a a
N , y
a S Z x
e -b
21. S ae,0
b P
P x, y
N
A’ A
-a a
N , y
a S Z x
e -b
SP ePN
22. S ae,0
b P
P x, y
N
A’ A
-a a
N , y
a S Z x
e -b
SP ePN
2
x ae 2 y 02 e x y y 2
a
e
2
2 a
x ae y e x
2 2
e
23. S ae,0
b P
P x, y
N
A’ A
-a a
N , y
a S Z x
e -b
SP ePN
2
x ae 2 y 02 e x y y 2
a
e
2
2 a
x ae y e x
2 2
e
x 2 2aex a 2 e 2 y 2 e 2 x 2 2aex a 2
x 2 1 e 2 y 2 a 2 1 e 2
24. S ae,0
b P
P x, y
N
A’ A
-a a
N , y
a S Z x
e -b
SP ePN
2
x ae 2 y 02 e x y y 2
a
e
2
2 a
x ae y e x
2 2
e
x 2 2aex a 2 e 2 y 2 e 2 x 2 2aex a 2
x 2 1 e 2 y 2 a 2 1 e 2
x2 y2
2 1
a a 1 e
2 2
25. b2
when x 0, y b 1
a 1 e
i.e. 2 2
b 2 a 2 1 e 2
26. b2
when x 0, y b 1
a 1 e
i.e. 2 2
b 2 a 2 1 e 2
Ellipse: (a > b) x2 y2
2
2 1
a b
where; b 2 a 2 1 e 2
focus : ae,0
a
directrices : x
e
e is the eccentricity
major semi-axis = a units
minor semi-axis = b units
27. b2
when x 0, y b 1
a 1 e
i.e. 2 2
b 2 a 2 1 e 2
Ellipse: (a > b) x2 y2 Note: If b > a
2
2 1
a b foci on the y axis
where; b a 1 e
2 2 2
a 2 b 2 1 e 2
focus : ae,0 focus : 0,be
a b
directrices : x directrices : y
e e
e is the eccentricity
major semi-axis = a units
minor semi-axis = b units
28. b2
when x 0, y b 1
a 1 e
i.e. 2 2
b 2 a 2 1 e 2
Ellipse: (a > b) x2 y2 Note: If b > a
2
2 1
a b foci on the y axis
where; b a 1 e
2 2 2
a 2 b 2 1 e 2
focus : ae,0 focus : 0,be
a b
directrices : x directrices : y
e e
e is the eccentricity
major semi-axis = a units Area ab
minor semi-axis = b units
29. e.g. Find the eccentricity, foci and directrices of the ellipse
x2 y2
1 and sketch the ellipse showing all of the important
9 5
features.
30. e.g. Find the eccentricity, foci and directrices of the ellipse
x2 y2
1 and sketch the ellipse showing all of the important
9 5
features.
x2 y2
1
9 5
a2 9
a3
31. e.g. Find the eccentricity, foci and directrices of the ellipse
x2 y2
1 and sketch the ellipse showing all of the important
9 5
features.
x2 y2
1 b2 5
9 5
a 2 1 e 2 5
a2 9
a3
32. e.g. Find the eccentricity, foci and directrices of the ellipse
x2 y2
1 and sketch the ellipse showing all of the important
9 5
features.
x2 y2
1 b2 5
9 5
a 2 1 e 2 5
91 e 2 5
a2 9
a3
5
1 e 2
9
4
e
2
9
2
e
3
33. e.g. Find the eccentricity, foci and directrices of the ellipse
x2 y2
1 and sketch the ellipse showing all of the important
9 5
features.
x2 y2
1 b2 5
9 5
a 2 1 e 2 5
91 e 2 5
a2 9
2
a3 eccentricity
5 3
1 e 2
9 foci : 2,0
4
e
2
3
9 directrices : x 3
2
2
e 9
3 x
2
47. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
48. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
49. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22
1
4 9
50. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22
1
4 9
centre : (1,2)
51. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22
1
4 9
centre : (1,2)
b2 9
b3
52. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22
1
4 9
centre : (1,2)
b2 9 a 2 b 2 1 e 2
b3 4 91 e 2
5
e
3
53. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22
1
4 9
centre : (1,2)
b2 9 a 2 b 2 1 e 2
b3 4 91 e 2
5
e
3
foci : 1,2 5
9
directrices : y 2
5
54. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22 Exercise 6A; 1, 2, 3, 5, 7,
1
4 9 8, 9, 11, 13, 15
centre : (1,2)
b2 9 a 2 b 2 1 e 2
b3 4 91 e 2
5
e
3
foci : 1,2 5
9
directrices : y 2
5