X2 t03 05 rectangular hyperbola (2013)

1. Rectangular Hyperbola

2. Rectangular Hyperbola
A hyperbola whose asymptotes are perpendicular to each other

3. Rectangular Hyperbola
A hyperbola whose asymptotes are perpendicular to each other
1


a
b
a
b

4. Rectangular Hyperbola
A hyperbola whose asymptotes are perpendicular to each other
1


a
b
a
b
a b
a b

2 2

5. Rectangular Hyperbola
A hyperbola whose asymptotes are perpendicular to each other
1



a
b
a
b
a b
a b

2 2equation; the has hyperbola

2 2 2
2
2
2
2
1
a y x
a
y
a
x
 
 

6. Rectangular Hyperbola
A hyperbola whose asymptotes are perpendicular to each other
1


a
b
a
b
a b
a b

2 2equation; the has hyperbola

2 2 2
2
2
2
2
1
a y x
a
y
a
x
 
 
22
2
2
aa
e
a



7. Rectangular Hyperbola
A hyperbola whose asymptotes are perpendicular to each other
1



a
b
a
b
a b
a b

2 2equation; the has hyperbola

2 2 2
2
2
2
2
1 a y x
a
y
a
x
 
 
22
2
2
aa
e
a


2
2
2
e
e



8. Rectangular Hyperbola
A hyperbola whose asymptotes are perpendicular to each other
1


a
b
a
b
a b
a b

2 2equation; the has hyperbola

2 2 2
2
2
2
2
1
a y x
a
y
a
x
 
 
2 isty eccentrici

22
2
2
aa
e
a


2
2
2
e
e



9. y
x
Y
X

y
x
P
,

10. y
x
Y
X

y
x
P
,
In order to make the
asymptotes the coordinate
axes we need to rotate the
curve 45 degrees
anticlockwise.

11. y
x
Y
X

y
x
P
,
In order to make the
asymptotes the coordinate
axes we need to rotate the
curve 45 degrees
anticlockwise.



45 cis by multiplied is , i.e.iy x yxP

12. y
x
Y
X

y
x
P
,
In order to make the
asymptotes the coordinate
axes we need to rotate the
curve 45 degrees
anticlockwise.



45 cis by multiplied is , i.e.iy x yxP



 
45 sin 45 cosi iy x





  
2
1
2
1
i iy x

13. y
x
Y
X

y
x
P
,
In order to make the
asymptotes the coordinate
axes we need to rotate the
curve 45 degrees
anticlockwise.



45 cis by multiplied is , i.e.iy x yxP



 
45 sin 45 cosi iy x





  
2
1
2
1
i iy x
 
i
yx yx
y iy ix x
i iy x
2 2
2
1
1
2
1




   
  

14. y
x
Y
X

y
x
P
,
In order to make the
asymptotes the coordinate
axes we need to rotate the
curve 45 degrees
anticlockwise.



45 cis by multiplied is , i.e.iy x yxP



 
45 sin 45 cosi iy x





  
2
1
2
1
i iy x
 
i
yx yx
y iy ix x
i iy x
2 2
2
1
1
2
1




   
  
2

2
y
x
Y
y
x
X






15. y
x
Y
X

y
x
P
,
In order to make the
asymptotes the coordinate
axes we need to rotate the
curve 45 degrees
anticlockwise.



45 cis by multiplied is , i.e.iy x yxP



 
45 sin 45 cosi iy x






  
2
1
2
1
i iy x
 
i
yx yx
y iy ix x
i iy x
2 2
2
1
1
2
1




   
  
2

2
y
x
Y
y
x
X





2
2 2
y x
XY



16. y
x
Y
X

y
x
P
,
In order to make the
asymptotes the coordinate
axes we need to rotate the
curve 45 degrees
anticlockwise.



45 cis by multiplied is , i.e.iy x yxP



 
45 sin 45 cosi iy x






  
2
1
2
1
i iy x
 
i
yx yx
y iy ix x
i iy x
2 2
2
1
1
2
1




   
  
2

2
y
x
Y
y
x
X





2
2 2
y x
XY


2
2
a
XY

17. focus;


0,ae


0,2a 


18. focus;


0,ae


0,2a 

ai a
i a








2
1
2
1
2

19. focus;


0,ae


0,2a 

ai a
i a








2
1
2
1
2


aa, focus


20. focus;


0,ae


0,2a 

ai a
i a








2
1
2
1
2


aa, focus

directrix;
e
a
x2
a
x

21. focus;


0,ae


0,2a 

ai a
i a








2
1
2
1
2


aa, focus

directrix;
e
a
x2
a
x
x y
y


to|| rotated when
axis to|| are s directrice

22. focus;


0,ae


0,2a 

ai a
i a








2
1
2
1
2


aa, focus

directrix;
e
a
x2
a
x
x y
y


to|| rotated when
axis to|| are s directrice
0 form in thus



k
y
x

23. focus;


0,ae


0,2a 

ai a
i a








2
1
2
1
2


aa, focus

directrix;
e
a
x2
a
x
x y
y


to|| rotated when
axis to|| are s directrice
0 form in thus



k
y
x
2
2
is s directrice between distance Now
a

24. focus;


0,ae


0,2a 

ai a
i a








2
1
2
1
2


aa, focus

directrix;
e
a
x2
a
x
x y
y


to|| rotated when
axis to|| are s directrice
0 form in thus



k
y
x
2
2
is s directrice between distance Now
a
2
is directrix origin to from distance
a


25. focus;


0,ae


0,2a 

ai a
i a








2
1
2
1
2


aa, focus

directrix;
e
a
x2
a
x
x y
y


to|| rotated when
axis to|| are s directrice
0 form in thus



k
y
x
2
2
is s directrice between distance Now
a
2
is directrix origin to from distance
a

2 2
00a k




26. focus;


0,ae


0,2a 

ai a
i a








2
1
2
1
2


aa, focus

directrix;
e
a
x2
a
x
x y
y


to|| rotated when
axis to|| are s directrice
0 form in thus



k
y
x
2
2
is s directrice between distance Now
a
2
is directrix origin to from distance
a

2 2
00a k



a
k
a k





27. focus;


0,ae


0,2a 

ai a
i a








2
1
2
1
2


aa, focus

directrix;
e
a
x2
a
x
x y
y


to|| rotated when
axis to|| are s directrice
0 form in thus



k
y
x
2
2
is s directrice between distance Now
a
2
is directrix origin to from distance
a

2 2
00a k



a
k
a k




a y
x




are s directrice

28. The rectangular hyperbola with x and yaxes as aymptotes,
has the equation;
2
2
1
a xy
where;

aa


, : foci
a y
x



:s directrice
2 ty eccentrici


29. The rectangular hyperbola with x and yaxes as aymptotes,
has the equation;
2
2
1
a xy
where;

aa


, : foci
a y
x



:s directrice
2 ty eccentrici

2
c xy ofs Coordinate Parametric


30. The rectangular hyperbola with x and yaxes as aymptotes,
has the equation;
2
2
1
a xy
where;

aa


, : foci
a y
x



:s directrice
2 ty eccentrici

2
c xy ofs Coordinate Parametric

c
t
x

tc
y

31. The rectangular hyperbola with
x
and
y
axes as aymptotes,
has the equation;
2
2
1
a xy
where
;

aa


, : foci
a y
x



:s directrice
2 ty eccentrici

2

c xy ofs Coordinate Parametric

c
t
x

tc
y
Tangent:ct ytx
2
2



32. The rectangular hyperbola with
x
and
y
axes as aymptotes,
has the equation;
2
2
1
a xy
where
;

aa


, : foci
a y
x



:s directrice
2 ty eccentrici

2

c xy ofs Coordinate Parametric

c
t
x

tc
y
Tangent:ct ytx
2
2


Normal:


1
4 3



tc ty xt

33. e.g. (
i
) (1991)
The hyperbola
H
is
xy
= 4
a) Sketch
H
showing where
H
intersects the axis of symmetry.

34. e.g. (
i
) (1991)
The hyperbola
H
is
xy
= 4
a) Sketch
H
showing where
H
intersects the axis of symmetry.
y
x
y
=
x

35. e.g. (
i
) (1991)
The hyperbola
H
is
xy
= 4
a) Sketch
H
showing where
H
intersects the axis of symmetry.
y
x
y
=
x
2
4
4
2


x
xx
y


2,2

2,2



36. e.g. (i) (1991)
The hyperbola His xy= 4
a) Sketch Hshowing where Hintersects the axis of symmetry.
y
x
y= x
2
4
4
2


x
xx
y


2,2

2,2


t ytx
t
t P4 is
2
,2 at tangent that the Show b)
2
  





37. e.g. (i) (1991)
The hyperbola His xy= 4
a) Sketch Hshowing where Hintersects the axis of symmetry.
y
x
y= x
2
4
4
2


x
xx
y


2,2

2,2


t ytx
t
t P4 is
2
,2 at tangent that the Show b)
2
  




2
4
4
x
dx
dy
x
y



38. e.g. (i) (1991)
The hyperbola His xy= 4
a) Sketch Hshowing where Hintersects the axis of symmetry.
y
x
y= x
2
4
4
2


x
xx
y


2,2

2,2


t ytx
t
t P4 is
2
,2 at tangent that the Show b)
2
  




2
4
4
x
dx
dy
x
y


 2
21

2
4
,2 when
t
t dx
dy
t x


 

39. e.g. (i) (1991)
The hyperbola His xy= 4
a) Sketch Hshowing where Hintersects the axis of symmetry.
y
x
y= x
2
4
4
2


x
xx
y


2,2

2,2


t ytx
t
t P4 is
2
,2 at tangent that the Show b)
2
  




2
4
4
x
dx
dy
x
y


 2
21

2
4
,2 when
t
t dx
dy
t x


 

t x
t
t
y2
1 2
2
  

40. e.g. (i) (1991)
The hyperbola His xy= 4
a) Sketch Hshowing where Hintersects the axis of symmetry.
y
x
y= x
2
4
4
2


x
xx
y


2,2

2,2


t ytx
t
t P4 is
2
,2 at tangent that the Show b)
2
  




2
4
4
x
dx
dy
x
y


 2
21

2
4
,2 when
t
t dx
dy
t x


 

t x
t
t
y2
1 2
2
  
t ytx
t x t yt
4
2 2
2
2
 





41. 



 






 
tsts
st
M
s
s Q P t s s
4
,
4
at intersect
2
,2 and at tangents that the show , ,0 c)
2 2

42. 



 






 
tsts
st
M
s
s Q P t s s
4
,
4
at intersect
2
,2 and at tangents that the show , ,0 c)
2 2
t ytxP4 :
2


s ysxQ4 :
2



43. 



 






 
tsts
st
M
s
s Q P t s s
4
,
4
at intersect
2
,2 and at tangents that the show , ,0 c)
2 2
t ytxP4 :
2


s ysxQ4 :
2




s t ys t4 4
2 2




44. 



 






 
tsts
st
M
s
s Q P t s s
4
,
4
at intersect
2
,2 and at tangents that the show , ,0 c)
2 2
t ytxP4 :
2


s ysxQ4 :
2




s t ys t4 4
2 2









t
s
y
s
t
ys
t
s
t






4
4

45. 



 






 
tsts
st
M
s
s Q P t s s
4
,
4
at intersect
2
,2 and at tangents that the show , ,0 c)
2 2
t ytxP4 :
2


s ysxQ4 :
2




s t ys t4 4
2 2









t
s
y
s
t
ys
t
s
t






4
4
t
t
s
t
x4
4
2




46. 



 






 
tsts
st
M
s
s Q P t s s
4
,
4
at intersect
2
,2 and at tangents that the show , ,0 c)
2 2
t ytxP4 :
2


s ysxQ4 :
2




s t ys t4 4
2 2









t
s
y
s
t
ys
t
s
t






4
4
t
t
s
t
x4
4
2



t
s
st
ts
t t st
x






4
4 4 4
2 2

47. 



 






 
tsts
st
M
s
s Q P t s s
4
,
4
at intersect
2
,2 and at tangents that the show , ,0 c)
2 2
t ytxP4 :
2


s ysxQ4 :
2




s t ys t4 4
2 2









t
s
y
s
t
ys
t
s
t






4
4
t
t
s
t
x4
4
2



t
s
st
ts
t t st
x






4
4 4 4
2 2




 

tsts
st
M
4
,
4
is

48. origin. the including not but origin, he through t line
straight a is of locus that the show ,
1
that Suppose d)M
t
s



49. origin. the including not but origin, he through t line
straight a is of locus that the show ,
1
that Suppose d)M
t
s


t
s
y
t
s
s
t
x




4

4

50. origin. the including not but origin, he through t line
straight a is of locus that the show ,
1
that Suppose d)M
t
s


t
s
y
t
s
s
t
x




4

4
t
s
1


1 st



51. origin. the including not but origin, he through t line
straight a is of locus that the show ,
1
that Suppose d)M
t
s


t
s
y
t
s
s
t
x




4

4
t
s
1


4
x
st



1 st



52. origin. the including not but origin, he through t line
straight a is of locus that the show ,
1
that Suppose d)M
t
s


t
s
y
t
s
s
t
x




4

4
t
s
1


4
x
st



4
y
st
x




1 st



53. origin. the including not but origin, he through t line
straight a is of locus that the show ,
1
that Suppose d)M
t
s


t
s
y
t
s
s
t
x




4

4
t
s
1


4
x
st



4
y
st
x




x
y



1 st



54. origin. the including not but origin, he through t line
straight a is of locus that the show ,
1
that Suppose d)
M
t
s


t
s
y
t
s
s
t
x




4

4
t
s
1


4
x
st



4
y
st
x




x
y




0,0 thus ,0
4
 

M
t
s
1 st



55. origin. the including not but origin, he through t line
straight a is of locus that the show ,
1
that Suppose d)
M
t
s


t
s
y
t
s
s
t
x




4

4
t
s
1


4
x
st



4
y
st
x




x
y




0,0 thus ,0
4
 

M
t
s


0,0 excluding , is of locus
x
y
M



1 st



56. (
ii
)
a S
P
P
S2 that Show



x


y
x
P
,
S
S


57. (ii)a S
P
P
S2 that Show



y
x


y
x
P
,
S
S

By definition of an ellipse;

58. (ii)a S
P
P
S2 that Show



y
x


y
x
P
,
S
S

By definition of an ellipse;
ePM S
P
P
S

e
a
x
M
e
a
x

59. (ii)a S
P
P
S2 that Show



y
x


y
x
P
,
S
S

By definition of an ellipse;
ePM S
P
P
S

M
eP


e
a
x
M
e
a
x
M


60. (ii)a S
P
P
S2 that Show



y
x


y
x
P
,
S
S

By definition of an ellipse;
ePM S
P
P
S

M
eP


e
a
x
M
e
a
x
M


M
P
P
Me



a
e
a
e
2
2









61. (ii)a S
P
P
S2 that Show



y
x


y
x
P
,
S
S

By definition of an ellipse;
ePM S
P
P
S

M
eP


e
a
x
M
e
a
x
M


M
P
P
Me



a
e
a
e
2
2








Exercise 6D; 3, 4, 7, 10, 11a,
12, 14, 19, 21, 26, 29,
31, 43, 47