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Rectangular Hyperbola
Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other
Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other
1
ab
ab
Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other
1
ab
ab
abab
22
Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other
1
ab
ab
abab
22
equation;thehas hyperbola
222
2
2
2
2
1
ayxay
ax
Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other
1
ab
ab
abab
22
222 1 aea equation;thehas hyperbola
222
2
2
2
2
1
ayxay
ax
Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other
1
ab
ab
abab
22
222 1 aea equation;thehas hyperbola
222
2
2
2
2
1
ayxay
ax
2211
2
2
ee
e
Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other
1
ab
ab
abab
22
222 1 aea equation;thehas hyperbola
222
2
2
2
2
1
ayxay
ax
2211
2
2
ee
e
2 isty eccentrici
y
x
Y
X
yxP ,
y
x
Y
X
yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.
y
x
Y
X
yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.
45cisby multiplied is , i.e. iyxyxP
y
x
Y
X
yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.
45cisby multiplied is , i.e. iyxyxP 45sin45cos iiyx
21
21 iiyx
y
x
Y
X
yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.
45cisby multiplied is , i.e. iyxyxP 45sin45cos iiyx
21
21 iiyx
iyxyx
yiyixx
iiyx
22
21
12
1
y
x
Y
X
yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.
45cisby multiplied is , i.e. iyxyxP 45sin45cos iiyx
21
21 iiyx
iyxyx
yiyixx
iiyx
22
21
12
1
2
2yxYyxX
y
x
Y
X
yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.
45cisby multiplied is , i.e. iyxyxP 45sin45cos iiyx
21
21 iiyx
iyxyx
yiyixx
iiyx
22
21
12
1
2
2yxYyxX
2
22 yxXY
y
x
Y
X
yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.
45cisby multiplied is , i.e. iyxyxP 45sin45cos iiyx
21
21 iiyx
iyxyx
yiyixx
iiyx
22
21
12
1
2
2yxYyxX
2
22 yxXY
2
2aXY
focus; 0,ae
0,2a
focus; 0,ae
0,2a
aia
ia
21
212
focus; 0,ae
0,2a
aia
ia
21
212
aa,focus
focus; 0,ae
0,2a
aia
ia
21
212
aa,focus
directrix;eax
2ax
focus; 0,ae
0,2a
aia
ia
21
212
aa,focus
directrix;eax
2ax
xyy
to|| rotatedwhen axisto||aresdirectrice
focus; 0,ae
0,2a
aia
ia
21
212
aa,focus
directrix;eax
2ax
xyy
to|| rotatedwhen axisto||aresdirectrice
0formin thus kyx
focus; 0,ae
0,2a
aia
ia
21
212
aa,focus
directrix;eax
2ax
xyy
to|| rotatedwhen axisto||aresdirectrice
0formin thus kyx
22 is sdirectricebetween distance Now a
focus; 0,ae
0,2a
aia
ia
21
212
aa,focus
directrix;eax
2ax
xyy
to|| rotatedwhen axisto||aresdirectrice
0formin thus kyx
22 is sdirectricebetween distance Now a
2 isdirectrix origin to from distance a
focus; 0,ae
0,2a
aia
ia
21
212
aa,focus
directrix;eax
2ax
xyy
to|| rotatedwhen axisto||aresdirectrice
0formin thus kyx
22 is sdirectricebetween distance Now a
2 isdirectrix origin to from distance a
2200 ak
focus; 0,ae
0,2a
aia
ia
21
212
aa,focus
directrix;eax
2ax
xyy
to|| rotatedwhen axisto||aresdirectrice
0formin thus kyx
22 is sdirectricebetween distance Now a
2 isdirectrix origin to from distance a
2200 ak
akak
focus; 0,ae
0,2a
aia
ia
21
212
aa,focus
directrix;eax
2ax
xyy
to|| rotatedwhen axisto||aresdirectrice
0formin thus kyx
22 is sdirectricebetween distance Now a
2 isdirectrix origin to from distance a
2200 ak
akak
ayx aresdirectrice
The rectangular hyperbola with x and y axes as aymptotes, has the equation;
2
21 axy
where; aa , :foci
ayx :sdirectrice
2ty eccentrici
The rectangular hyperbola with x and y axes as aymptotes, has the equation;
2
21 axy
where; aa , :foci
ayx :sdirectrice
2ty eccentrici
2 cxyofsCoordinateParametric
The rectangular hyperbola with x and y axes as aymptotes, has the equation;
2
21 axy
where; aa , :foci
ayx :sdirectrice
2ty eccentrici
2 cxyofsCoordinateParametric
ctx tcy
The rectangular hyperbola with x and y axes as aymptotes, has the equation;
2
21 axy
where; aa , :foci
ayx :sdirectrice
2ty eccentrici
2 cxyofsCoordinateParametric
ctx tcy
Tangent: ctytx 22
The rectangular hyperbola with x and y axes as aymptotes, has the equation;
2
21 axy
where; aa , :foci
ayx :sdirectrice
2ty eccentrici
2 cxyofsCoordinateParametric
ctx tcy
Tangent: ctytx 22 Normal: 143 tctyxt
e.g. (i) (1991)The hyperbola H is xy= 4
a) Sketch H showing where H intersects the axis of symmetry.
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
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
244
2
xxxy
2,2
2,2
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
244
2
xxxy
2,2
2,2
tytxt
tP 4 is 2,2at tangent that theShow b) 2
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
244
2
xxxy
2,2
2,2
tytxt
tP 4 is 2,2at tangent that theShow b) 2
24
4
xdxdy
xy
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
244
2
xxxy
2,2
2,2
tytxt
tP 4 is 2,2at tangent that theShow b) 2
24
4
xdxdy
xy
2
2
1
24,2when
t
tdxdytx
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
244
2
xxxy
2,2
2,2
tytxt
tP 4 is 2,2at tangent that theShow b) 2
24
4
xdxdy
xy
2
2
1
24,2when
t
tdxdytx
txtt
y 2122
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
244
2
xxxy
2,2
2,2
tytxt
tP 4 is 2,2at tangent that theShow b) 2
24
4
xdxdy
xy
2
2
1
24,2when
t
tdxdytx
txtt
y 2122
tytxtxtyt
422
2
2
tstsstM
ssQPtss
4,4at intersect
2,2 and at tangents that theshow ,,0 c) 22
tstsstM
ssQPtss
4,4at intersect
2,2 and at tangents that theshow ,,0 c) 22
tytxP 4: 2 sysxQ 4: 2
tstsstM
ssQPtss
4,4at intersect
2,2 and at tangents that theshow ,,0 c) 22
tytxP 4: 2 sysxQ 4: 2
styst 4422
tstsstM
ssQPtss
4,4at intersect
2,2 and at tangents that theshow ,,0 c) 22
tytxP 4: 2 sysxQ 4: 2
styst 4422
tsy
stystst
4
4
tstsstM
ssQPtss
4,4at intersect
2,2 and at tangents that theshow ,,0 c) 22
tytxP 4: 2 sysxQ 4: 2
styst 4422
tsy
stystst
4
4
tts
tx 44 2
tstsstM
ssQPtss
4,4at intersect
2,2 and at tangents that theshow ,,0 c) 22
tytxP 4: 2 sysxQ 4: 2
styst 4422
tsy
stystst
4
4
tts
tx 44 2
tsst
tsttstx
4
444 22
tstsstM
ssQPtss
4,4at intersect
2,2 and at tangents that theshow ,,0 c) 22
tytxP 4: 2 sysxQ 4: 2
styst 4422
tsy
stystst
4
4
tts
tx 44 2
tsst
tsttstx
4
444 22
tstsstM 4,4 is
origin. theincludingnot but origin, he through tline
straight a is of locus that theshow,1 that Suppose d) Mt
s
origin. theincludingnot but origin, he through tline
straight a is of locus that theshow,1 that Suppose d) Mt
s
tsy
tsstx
4 4
origin. theincludingnot but origin, he through tline
straight a is of locus that theshow,1 that Suppose d) Mt
s
tsy
tsstx
4 4
ts 1
1st
origin. theincludingnot but origin, he through tline
straight a is of locus that theshow,1 that Suppose d) Mt
s
tsy
tsstx
4 4
ts 1
4xs t
1st
origin. theincludingnot but origin, he through tline
straight a is of locus that theshow,1 that Suppose d) Mt
s
tsy
tsstx
4 4
ts 1
4xs t
4ys t
x
1st
origin. theincludingnot but origin, he through tline
straight a is of locus that theshow,1 that Suppose d) Mt
s
tsy
tsstx
4 4
ts 1
4xs t
4ys t
x
xy
1st
origin. theincludingnot but origin, he through tline
straight a is of locus that theshow,1 that Suppose d) Mt
s
tsy
tsstx
4 4
ts 1
4xs t
4ys t
x
xy 0,0 thus,04
M
ts
1st
origin. theincludingnot but origin, he through tline
straight a is of locus that theshow,1 that Suppose d) Mt
s
tsy
tsstx
4 4
ts 1
4xs t
4ys t
x
xy 0,0 thus,04
M
ts 0,0excluding,isoflocus xyM
1st
(ii) aSPPS 2 that Show
x
yxP ,
SS
(ii) aSPPS 2 that Show
y
x
yxP ,
SS
By definition of an ellipse;
(ii) aSPPS 2 that Show
y
x
yxP ,
SS
By definition of an ellipse;
ePMSPPS
eax
M
eax
(ii) aSPPS 2 that Show
y
x
yxP ,
SS
By definition of an ellipse;
ePMSPPS MeP
eax
M
eax
M
(ii) aSPPS 2 that Show
y
x
yxP ,
SS
By definition of an ellipse;
ePMSPPS MeP
eax
M
eax
M
MPPMe
aeae
2
2
(ii) aSPPS 2 that Show y
x
yxP ,
SS
By definition of an ellipse;
ePMSPPS MeP
eax
M
eax
M
MPPMe
aeae
2
2
Exercise 6D; 3, 4, 7, 10, 11a,12, 14, 19, 21, 26, 29,
31, 43, 47
Komarami Unit 3
