x2 t03 06 chord of contact and properties (2010)
TRANSCRIPT
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Chord of Contacty
xa-a
b
-b
00 , yxT
From an external point T, two tangents may be drawn.
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Chord of Contact
1equation has at tangent 21
21
byy
axxP
y
xa-a
b
-b
11, yxP 00 , yxT
From an external point T, two tangents may be drawn.
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Chord of Contact
1equation has at tangent 21
21
byy
axxP
y
xa-a
b
-b
11, yxP
1equation has at tangent 22
22
byy
axxQ
22 , yxQ
00 , yxT
From an external point T, two tangents may be drawn.
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Chord of Contact
1equation has at tangent 21
21
byy
axxP
y
xa-a
b
-b
11, yxP
1equation has at tangent 22
22
byy
axxQ
22 , yxQ
Now T
00 , yxT
lies on both lines,
From an external point T, two tangents may be drawn.
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Chord of Contact
1equation has at tangent 21
21
byy
axxP
y
xa-a
b
-b
11, yxP
1equation has at tangent 22
22
byy
axxQ
22 , yxQ
Now T
00 , yxT
lies on both lines,1 and 1 2
022
022
012
01 b
yyaxx
byy
axx
From an external point T, two tangents may be drawn.
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y
xa-a
b
-b
11, yxP
22 , yxQ
00 , yxT
1201
201
byy
axx
1202
202
byy
axx
Thus P and Q both must lie on a line with equation;
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y
xa-a
b
-b
11, yxP
22 , yxQ
00 , yxT
1201
201
byy
axx
1202
202
byy
axx
Thus P and Q both must lie on a line with equation;
120
20
byy
axx
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y
xa-a
b
-b
11, yxP
22 , yxQ
00 , yxT
1201
201
byy
axx
1202
202
byy
axx
Thus P and Q both must lie on a line with equation;
120
20
byy
axx
which must be the line PQ i.e. chord of contact
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y
xa-a
b
-b
11, yxP
22 , yxQ
00 , yxT
1201
201
byy
axx
1202
202
byy
axx
Thus P and Q both must lie on a line with equation;
120
20
byy
axx
which must be the line PQ i.e. chord of contact
Similarly the chord of contact of the hyperbola has the equation;
120
20
byy
axx
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Rectangular Hyperbola y
x2cxy
1) Show that equation PQ
pccpP ,
qccqQ ,
)1(is qpcpqyx
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Rectangular Hyperbola y
x2cxy
1) Show that equation PQ
pccpP ,
qccqQ ,
)1(is qpcpqyx
2) Show that T
00 , yxT
qpc
qpcpq 2,2 scoordinate has
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Rectangular Hyperbola y
x2cxy
1) Show that equation PQ
pccpP ,
qccqQ ,
)1(is qpcpqyx
2) Show that T
00 , yxT
qpc
qpcpq 2,2 scoordinate has
qpcy
qpcpqx
2 200
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Rectangular Hyperbola y
x2cxy
1) Show that equation PQ
pccpP ,
qccqQ ,
)1(is qpcpqyx
2) Show that T
00 , yxT
qpc
qpcpq 2,2 scoordinate has
qpcy
qpcpqx
2 200
Substituting into (1); qpcyc
xqpx
2
0
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Rectangular Hyperbola y
x2cxy
1) Show that equation PQ
pccpP ,
qccqQ ,
)1(is qpcpqyx
2) Show that T
00 , yxT
qpc
qpcpq 2,2 scoordinate has
qpcy
qpcpqx
2 200
Substituting into (1); qpcyc
xqpx
2
0
0
2
0
0 222
ycy
cycxx
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Rectangular Hyperbola y
x2cxy
1) Show that equation PQ
pccpP ,
qccqQ ,
)1(is qpcpqyx
2) Show that T
00 , yxT
qpc
qpcpq 2,2 scoordinate has
qpcy
qpcpqx
2 200
Substituting into (1); qpcyc
xqpx
2
0
0
2
0
0 222
ycy
cycxx 2
00 2cyxxy
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Geometric Properties
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Geometric Properties(1) The chord of contact from a point on the directrix is a focal
chord.
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Geometric Properties(1) The chord of contact from a point on the directrix is a focal
chord.
ellipse
0, scoordinate hasit directrix on the is As yeaT
eax 0 i.e.
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Geometric Properties(1) The chord of contact from a point on the directrix is a focal
chord.
ellipse
0, scoordinate hasit directrix on the is As yeaT
eax 0 i.e.
equation;thehavellcontact wi of chord
1
1
20
20
2
byy
aex
byy
aeax
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Geometric Properties(1) The chord of contact from a point on the directrix is a focal
chord.
ellipse
0, scoordinate hasit directrix on the is As yeaT
eax 0 i.e.
equation;thehavellcontact wi of chord
1
1
20
20
2
byy
aex
byy
aeax
Substitute in focus (ae,0)
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Geometric Properties(1) The chord of contact from a point on the directrix is a focal
chord.
ellipse
0, scoordinate hasit directrix on the is As yeaT
eax 0 i.e.
equation;thehavellcontact wi of chord
1
1
20
20
2
byy
aex
byy
aeax
Substitute in focus (ae,0)
1
010
aeae
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Geometric Properties(1) The chord of contact from a point on the directrix is a focal
chord.
ellipse
0, scoordinate hasit directrix on the is As yeaT
eax 0 i.e.
equation;thehavellcontact wi of chord
1
1
20
20
2
byy
aex
byy
aeax
Substitute in focus (ae,0)
1
010
aeae
contactofchordon liesfocusi.e. it is a focal chord
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(2) That part of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.
y
x
sin,cos baPT
eax
S
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(2) That part of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.
y
x
sin,cos baPT
eax
SProve: 90PST
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(2) That part of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.
y
x
sin,cos baPT
eax
SProve: 90PST
1sincos is tangent ofequation b
ya
x
1sincos,when
by
aea
eax
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(2) That part of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.
y
x
sin,cos baPT
eax
SProve: 90PST
1sincos is tangent ofequation b
ya
x
1sincos,when
by
aea
eax
sincos
cossin
1sincos
eeby
ee
by
by
e
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(2) That part of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.
y
x
sin,cos baPT
eax
SProve: 90PST
1sincos is tangent ofequation b
ya
x
1sincos,when
by
aea
eax
sincos
cossin
1sincos
eeby
ee
by
by
e
sincos,
eeb
eaT
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aeabmPS
cos0sin
eab
cossin
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aeabmPS
cos0sin
eab
cossin
aeea
eeb
mTS
0
sincos
2sin
cosaeae
eeb
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aeabmPS
cos0sin
eab
cossin
aeea
eeb
mTS
0
sincos
2sin
cosaeae
eeb
sin1
cos2ea
eb
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aeabmPS
cos0sin
eab
cossin
aeea
eeb
mTS
0
sincos
2sin
cosaeae
eeb
sin1
cos2ea
eb
sin1cos22
aea
eb
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aeabmPS
cos0sin
eab
cossin
aeea
eeb
mTS
0
sincos
2sin
cosaeae
eeb
sin1
cos2ea
eb
sin1cos22
aea
eb
sincos
bea
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aeabmPS
cos0sin
eab
cossin
aeea
eeb
mTS
0
sincos
2sin
cosaeae
eeb
sin1
cos2ea
eb
sin1cos22
aea
eb
sincos
bea
1sin
coscos
sin
bea
eabmm TSPS
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aeabmPS
cos0sin
eab
cossin
aeea
eeb
mTS
0
sincos
2sin
cosaeae
eeb
sin1
cos2ea
eb
sin1cos22
aea
eb
sincos
bea
1sin
coscos
sin
bea
eabmm TSPS 90 PST
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(3) Reflection PropertyTangent to an ellipse at a point P on it is equally inclined to the focal chords through P.
y
x
PT
S
T
S
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(3) Reflection PropertyTangent to an ellipse at a point P on it is equally inclined to the focal chords through P.
y
x
PT
S
T
S
Prove: TPSSPT
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(3) Reflection PropertyTangent to an ellipse at a point P on it is equally inclined to the focal chords through P.
y
x
PT
S
T
S
Prove: TPSSPT Construct a line || y axis passing through P
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(3) Reflection PropertyTangent to an ellipse at a point P on it is equally inclined to the focal chords through P.
y
x
PT
S
T
S
Prove: TPSSPT Construct a line || y axis passing through P
NPPN
TPPT
(ratio of intercepts of || lines)
NN
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(3) Reflection PropertyTangent to an ellipse at a point P on it is equally inclined to the focal chords through P.
y
x
PT
S
T
S
Prove: TPSSPT Construct a line || y axis passing through P
NPPN
TPPT
(ratio of intercepts of || lines)
NPTP
PNPT
NN
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SPNePPSePN and
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SPNePPSePN and
eSPTP
ePSPT
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SPNePPSePN and
eSPTP
ePSPT
SPTP
PSPT
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SPNePPSePN and
eSPTP
ePSPT
SPTP
PSPT
90 TSPPST (proven in property (2))
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SPNePPSePN and
eSPTP
ePSPT
SPTP
PSPT
90 TSPPST (proven in property (2))
TPSSPT secsec
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SPNePPSePN and
eSPTP
ePSPT
SPTP
PSPT
90 TSPPST (proven in property (2))
TPSSPT secsec
TPSSPT
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SPNePPSePN and
eSPTP
ePSPT
SPTP
PSPT
90 TSPPST (proven in property (2))
TPSSPT secsec
TPSSPT
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e.g. Find the cartesian equation of 2 2 8z z
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e.g. Find the cartesian equation of 2 2 8z z
The sum of the focal lengths of an ellipse is constant
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e.g. Find the cartesian equation of 2 2 8z z
2 84
aa
The sum of the focal lengths of an ellipse is constant
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e.g. Find the cartesian equation of 2 2 8z z
2 84
aa
2ae
4 212
e
e
The sum of the focal lengths of an ellipse is constant
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e.g. Find the cartesian equation of 2 2 8z z
2 84
aa
2ae
4 212
e
e
2 2 21b a e
2 116 14
12
b
The sum of the focal lengths of an ellipse is constant
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e.g. Find the cartesian equation of 2 2 8z z
2 84
aa
2ae
4 212
e
e
2 2 21b a e
2 116 14
12
b
The sum of the focal lengths of an ellipse is constant
2 2
locus is the ellipse 116 12x y
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e.g. Find the cartesian equation of 2 2 8z z
2 84
aa
2ae
4 212
e
e
2 2 21b a e
2 116 14
12
b
Exercise 6E; 1, 2, 4, 7, 8, 10
The sum of the focal lengths of an ellipse is constant
2 2
locus is the ellipse 116 12x y