x2 t03 01 ellipse (2011)
TRANSCRIPT
Conics
ConicsThe locus of points whose distance from a fixed point (focus) is a multiple, e, (eccentricity) of its distance from a fixed line (directrix)
ConicsThe locus of points whose distance from a fixed point (focus) is a multiple, e, (eccentricity) of its distance from a fixed line (directrix)
ConicsThe 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
ConicsThe 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
ConicsThe 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
ConicsThe 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
Ellipse (e < 1)y
xa-a
b
-b
AA’
Ellipse (e < 1)y
xa-a
b
-b
AA’S Z
Ellipse (e < 1)y
xa-a
b
-b
AA’S Z
SA = eAZ and SA’ = eA’Z
Ellipse (e < 1)y
xa-a
b
-b
AA’S Z
SA = eAZ and SA’ = eA’Z
(1) SA’ + SA = 2a(2) SA’ – SA = e(A’Z – AZ)
Ellipse (e < 1)y
xa-a
b
-b
AA’S Z
SA = eAZ and SA’ = eA’Z
(1) SA’ + SA = 2a(2) SA’ – SA = e(A’Z – AZ)
= e(AA’)= e(2a)= 2ae
xa-a
b
-b
AA’S Z
(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)
xa-a
b
-b
AA’S Z
(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)
(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)
xa-a
b
-b
AA’S Z
OS = OA - SA
(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)
Focus
(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)
xa-a
b
-b
AA’S Z
OS = OA - SA
(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)
Focus
= a – a(1 – e)= ae
(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)
0,aeS
xa-a
b
-b
AA’S Z
OS = OA - SA
(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)
Focus
= a – a(1 – e)= ae 0,aeS
OZ = OA + AZ
Directrix
(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)
xa-a
b
-b
AA’S Z
OS = OA - SA
(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)
Focus
= a – a(1 – e)= ae 0,aeS
OZ = OA + AZ
Directrix
eSAOA
(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)
SA eAZ
xa-a
b
-b
AA’S Z
OS = OA - SA
(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)
Focus
= a – a(1 – e)= ae 0,aeS
OZ = OA + AZ
Directrix
eax sdirectrice
eSAOA
(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)
ea
eea
eae
1
SA eAZ
xa-a
b
-b
PAA’
S Z
0,aeS
yxP , N
y
eaN ,
xa-a
b
-b
PAA’
S Z
0,aeS
yxP , N
y
eaN ,
ePNSP
xa-a
b
-b
PAA’
S Z
0,aeS
yxP , N
y
eaN ,
ePNSP
2
222
22
22 0
eaxeyaex
yyeaxeyaex
xa-a
b
-b
PAA’
S Z
0,aeS
yxP , N
y
eaN ,
ePNSP
2
222
22
22 0
eaxeyaex
yyeaxeyaex
22222
2222222
1122eayex
aaexxeyeaaexx
xa-a
b
-b
PAA’
S Z
0,aeS
yxP , N
y
eaN ,
ePNSP
2
222
22
22 0
eaxeyaex
yyeaxeyaex
22222
2222222
1122eayex
aaexxeyeaaexx
11 22
2
2
2
ea
yax
byx ,0when 222
22
2
1
11
i.e.
eabea
b
byx ,0when 222
22
2
1
11
i.e.
eabea
b
Ellipse: (a > b) 12
2
2
2
by
ax
222 1 where; eab
0,:focus ae
eax :sdirectrice
e is the eccentricity
major semi-axis = a units
minor semi-axis = b units
byx ,0when 222
22
2
1
11
i.e.
eabea
b
Ellipse: (a > b) 12
2
2
2
by
ax
222 1 where; eab
0,:focus ae
eax :sdirectrice
e is the eccentricity
major semi-axis = a units
minor semi-axis = b units
Note: If b > a
foci on the y axis 222 1 eba
be,0:focus
eby :sdirectrice
byx ,0when 222
22
2
1
11
i.e.
eabea
b
Ellipse: (a > b) 12
2
2
2
by
ax
222 1 where; eab
0,:focus ae
eax :sdirectrice
e is the eccentricity
major semi-axis = a units
minor semi-axis = b units
Note: If b > a
foci on the y axis 222 1 eba
be,0:focus
eby :sdirectrice
abArea
e.g.
features.
important theof all showing ellipse sketch the and 159
ellipsetheofsdirectriceandfocity,eccentrici theFind22
yx
e.g.
features.
important theof all showing ellipse sketch the and 159
ellipsetheofsdirectriceandfocity,eccentrici theFind22
yx
159
22
yx
392
aa
e.g.
features.
important theof all showing ellipse sketch the and 159
ellipsetheofsdirectriceandfocity,eccentrici theFind22
yx
159
22
yx
392
aa
52 b 51 22 ea
e.g.
features.
important theof all showing ellipse sketch the and 159
ellipsetheofsdirectriceandfocity,eccentrici theFind22
yx
159
22
yx
392
aa
52 b 51 22 ea
3294951
519
2
2
2
e
e
e
e
e.g.
features.
important theof all showing ellipse sketch the and 159
ellipsetheofsdirectriceandfocity,eccentrici theFind22
yx
159
22
yx
392
aa
52 b
0,2:foci
32ty eccentrici
51 22 ea
3294951
519
2
2
2
e
e
e
e
29
233:sdirectrice
x
x
y
x
Auxiliary circle
3-3
y
x
Auxiliary circle
3-3
35
ab
y
x
Auxiliary circle
3-3
35
ab
5
5
y
x
Auxiliary circle
3-3
35
ab
5
5
y
x
Auxiliary circle
3-3
35
ab
5
5
y
x
Auxiliary circle
3-3
35
ab
5
5
y
x
Auxiliary circle
3-3
35
ab
5
5
y
x
Auxiliary circle
3-3
35
ab
5
5
y
x
Auxiliary circle
3-3
35
ab
5
5
y
x
Auxiliary circle
3-3
35
ab
5
5
S(2,0)S’(-2,0)
y
x
Auxiliary circle
3-3
35
ab
5
5
S(2,0)S’(-2,0)
29
x 29
x
y
x
Auxiliary circle
3-3
35
ab
S(2,0)S’(-2,0)
5
5
29
x29
x
Major axis = 6 units unitsaxisMinor 52
(ii) 011161849 22 yxyx
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
94
41
3611
92
41 22
yx
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
94
41
3611
92
41 22
yx
19
241 22
yx
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
)2,1(:centre
94
41
3611
92
41 22
yx
19
241 22
yx
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
)2,1(:centre
392
bb
94
41
3611
92
41 22
yx
19
241 22
yx
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
)2,1(:centre
392
bb
35194
12
222
e
eeba
94
41
3611
92
41 22
yx
19
241 22
yx
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
)2,1(:centre
392
bb
35194
12
222
e
eeba
52,1 :foci 5
92 :sdirectrice y
94
41
3611
92
41 22
yx
19
241 22
yx
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
)2,1(:centre
392
bb
35194
12
222
e
eeba
52,1 :foci 5
92 :sdirectrice y
94
41
3611
92
41 22
yx
19
241 22
yx Exercise 6A; 1, 2, 3, 5, 7,
8, 9, 11, 13, 15