Conics & Parametersy
xa-a
a
-a
1) Circle
222 ayx
Conics & Parametersy
xa-a
a
-a
1) Circle
222 ayx
yxP ,
Conics & Parametersy
xa-a
a
-a
1) Circle
yxP ,a
x
222 ayx
cosax
Conics & Parametersy
xa-a
a
-a
1) Circle
yxP ,a
x
222 ayx
cosax
cosax
Conics & Parametersy
xa-a
a
-a
1) Circle
yxP ,a
x
cosax
cosax
y
sinay
sinay
222 ayx
Conics & Parametersy
xa-a
a
-a
1) Circle
yxP ,a
x
cosax
cosax
y
sinay
sinay
222 ayx
Proof:
2
222
222222
sincossincos
aa
aayx
2) Ellipse y
xa-a
b
-b
yxP ,
2) Ellipse y
xa-a
b
-b222 ayx
yxP ,
2) Ellipse y
xa-a
b
-b222 ayx
yxP ,
x
2) Ellipse y
xa-a
b
-b222 ayx
yxP ,
cosax
cosax
x
2) Ellipse y
xa-a
b
-b222 ayx
yxP ,
cosax
cosax
xy
2) Ellipse y
xa-a
b
-b222 ayx
yxP ,
cosax
cosax
xy
sinby
sinby
2) Ellipse y
xa-a
b
-b
yxP ,
xy
222 ayx
222 byx
cosax
cosax
sinby
sinby
Proof:
1sincos
sincos
22
2
22
2
22
2
2
2
2
b
ba
aby
ax
Equation of Tangent and Normaly
xa-a
b
-b
11, yxP
12
2
2
2
by
ax
Equation of Tangent and Normaly
xa-a
b
-b
11, yxP
12
2
2
2
by
ax
02222
dxdy
by
ax
dxdy
dydf
dxdf
Equation of Tangent and Normaly
xa-a
b
-b
11, yxP
12
2
2
2
by
ax
02222
dxdy
by
ax
dxdy
dydf
dxdf
yaxb
dxdy
ax
dxdy
by
2
2
2222
Equation of Tangent and Normaly
xa-a
b
-b
11, yxP
12
2
2
2
by
ax
02222
dxdy
by
ax
dxdy
dydf
dxdf
yaxb
dxdy
ax
dxdy
by
2
2
2222
11,at yxP
12
12
yaxb
dxdy
y
xa-a
b
-b
11, yxP
tangent:
11
21
2
1 xxyaxbyy
y
xa-a
b
-b
11, yxP
tangent:
11
21
2
1 xxyaxbyy
21
221
21
21
2
21
21
221
21
2
yaxbyyaxxb
xbxxbyayya
y
xa-a
b
-b
11, yxP
tangent:
11
21
2
1 xxyaxbyy
21
221
21
21
2
21
21
221
21
2
yaxbyyaxxb
xbxxbyayya
2
21
2
21
21
21
by
ax
byy
axx
y
xa-a
b
-b
11, yxP
tangent:
11
21
2
1 xxyaxbyy
21
221
21
21
2
21
21
221
21
2
yaxbyyaxxb
xbxxbyayya
2
21
2
21
21
21
by
ax
byy
axx
121
21
byy
axx
y
xa-a
b
-b
11, yxP
tangent:
11
21
2
1 xxyaxbyy
21
221
21
21
2
21
21
221
21
2
yaxbyyaxxb
xbxxbyayya
2
21
2
21
21
21
by
ax
byy
axx
121
21
byy
axx
normal:
11
21
2
1 xxxbyayy
y
xa-a
b
-b
11, yxP
tangent:
11
21
2
1 xxyaxbyy
21
221
21
21
2
21
21
221
21
2
yaxbyyaxxb
xbxxbyayya
2
21
2
21
21
21
by
ax
byy
axx
121
21
byy
axx
normal:
11
21
2
1 xxxbyayy
112
112
12
12
112
12
112
12
yxbyxayxbxya
yxaxyayxbyxb
y
xa-a
b
-b
11, yxP
tangent:
11
21
2
1 xxyaxbyy
21
221
21
21
2
21
21
221
21
2
yaxbyyaxxb
xbxxbyayya
2
21
2
21
21
21
by
ax
byy
axx
121
21
byy
axx
normal:
11
21
2
1 xxxbyayy
112
112
12
12
112
12
112
12
yxbyxayxbxya
yxaxyayxbyxb
22
1
2
1
2
bay
ybx
xa
Using Parametric Coordinates; sin,cosat baP
Using Parametric Coordinates; sin,cosat baP
sincos
sincos
2
2
ab
baab
dxdy
Using Parametric Coordinates; sin,cosat baP
sincos
sincos
2
2
ab
baab
dxdy
tangent:
cos
sincossin ax
abby
Using Parametric Coordinates; sin,cosat baP
sincos
sincos
2
2
ab
baab
dxdy
tangent:
cos
sincossin ax
abby
22
22
cossinsincoscoscossinsin
ababaybxabbxabay
Using Parametric Coordinates; sin,cosat baP
sincos
sincos
2
2
ab
baab
dxdy
tangent:
cos
sincossin ax
abby
22
22
cossinsincoscoscossinsin
ababaybxabbxabay
22 cossinsincos
by
ax
Using Parametric Coordinates; sin,cosat baP
sincos
sincos
2
2
ab
baab
dxdy
tangent:
cos
sincossin ax
abby
22
22
cossinsincoscoscossinsin
ababaybxabbxabay
22 cossinsincos
by
ax
1sincos
by
ax
Using Parametric Coordinates; sin,cosat baP
sincos
sincos
2
2
ab
baab
dxdy
tangent:
cos
sincossin ax
abby
22
22
cossinsincoscoscossinsin
ababaybxabbxabay
22 cossinsincos
by
ax
1sincos
by
ax
normal:
cos
cossinsin ax
baby
Using Parametric Coordinates; sin,cosat baP
sincos
sincos
2
2
ab
baab
dxdy
tangent:
cos
sincossin ax
abby
22
22
cossinsincoscoscossinsin
ababaybxabbxabay
22 cossinsincos
by
ax
1sincos
by
ax
normal:
cos
cossinsin ax
baby
cossin cossin
cossinsin cossincos
22
2
2
babyax
aaxbby
Using Parametric Coordinates; sin,cosat baP
sincos
sincos
2
2
ab
baab
dxdy
tangent:
cos
sincossin ax
abby
22
22
cossinsincoscoscossinsin
ababaybxabbxabay
22 cossinsincos
by
ax
1sincos
by
ax
normal:
cos
cossinsin ax
baby
cossin cossin
cossinsin cossincos
22
2
2
babyax
aaxbby
22
sincosbabyax
For ellipse 12
2
2
2
by
ax
11,at tangent yx
121
21
byy
axx
For ellipse 12
2
2
2
by
ax
11,at tangent yx
121
21
byy
axx
11,at normal yx
22
1
2
1
2
bay
ybx
xa
For ellipse 12
2
2
2
by
ax
11,at tangent yx
121
21
byy
axx
11,at normal yx
22
1
2
1
2
bay
ybx
xa
sin,cosat tangent ba
1sincos
by
ax
For ellipse 12
2
2
2
by
ax
11,at tangent yx
121
21
byy
axx
11,at normal yx
22
1
2
1
2
bay
ybx
xa
sin,cosat tangent ba sin,cosat normal ba
1sincos
by
ax 22
sincosbabyax
e.g. (i)
23,2point at the
116 ellipse theo tangent t theofequation theFind 2
2
yx
e.g. (i)
23,2point at the
116 ellipse theo tangent t theofequation theFind 2
2
yx
yx
dxdydxdyyx
16
028
e.g. (i)
23,2point at the
116 ellipse theo tangent t theofequation theFind 2
2
yx
yx
dxdydxdyyx
16
028
341
2316
2 ,2
3,2at
dxdy
dxdy
e.g. (i)
23,2point at the
116 ellipse theo tangent t theofequation theFind 2
2
yx
yx
dxdydxdyyx
16
028
341
2316
2 ,2
3,2at
dxdy
dxdy
234
123 xy
e.g. (i)
23,2point at the
116 ellipse theo tangent t theofequation theFind 2
2
yx
yx
dxdydxdyyx
16
028
341
2316
2 ,2
3,2at
dxdy
dxdy
234
123 xy
08342634
yxxy
(ii)
6point at the
sin,2cosellipsethetonormaltheofequation theFind
yx
(ii)
6point at the
sin,2cosellipsethetonormaltheofequation theFind
yx
xfy sin
xfxfdxdy cos
(ii)
6point at the
sin,2cosellipsethetonormaltheofequation theFind
yx
xfy sin
xfxfdxdy cos
xfy cos
xfxfdxdy sin
(ii)
6point at the
sin,2cosellipsethetonormaltheofequation theFind
yx
xfy sin
xfxfdxdy cos
xfy cos
xfxfdxdy sin
cos2x
sin2ddx
(ii)
6point at the
sin,2cosellipsethetonormaltheofequation theFind
yx
xfy sin
xfxfdxdy cos
xfy cos
xfxfdxdy sin
cos2x
sin2ddx
siny
cosddy
(ii)
6point at the
sin,2cosellipsethetonormaltheofequation theFind
yx
xfy sin
xfxfdxdy cos
xfy cos
xfxfdxdy sin
cos2x
sin2ddx
siny
cosddy
sin2cos
dxd
ddy
dxdy
(ii)
6point at the
sin,2cosellipsethetonormaltheofequation theFind
yx
xfy sin
xfxfdxdy cos
xfy cos
xfxfdxdy sin
cos2x
sin2ddx
siny
cosddy
sin2cos
dxd
ddy
dxdy
23
123
,6
at
dxdy
(ii)
6point at the
sin,2cosellipsethetonormaltheofequation theFind
yx
xfy sin
xfxfdxdy cos
xfy cos
xfxfdxdy sin
cos2x
sin2ddx
siny
cosddy
sin2cos
dxd
ddy
dxdy
23
123
,6
at
dxdy 3
32
21
xy
(ii)
6point at the
sin,2cosellipsethetonormaltheofequation theFind
yx
xfy sin
xfxfdxdy cos
xfy cos
xfxfdxdy sin
cos2x
sin2ddx
siny
cosddy
sin2cos
dxd
ddy
dxdy
23
123
,6
at
dxdy 3
32
21
xy
033324344332
yxxy
(iii)
22222
2
2
2
then ,1
ellipsetheotangent taisif that Show
kbmaby
ax
kmxy
(iii)
22222
2
2
2
then ,1
ellipsetheotangent taisif that Show
kbmaby
ax
kmxy
cosax
sinaddx
sinby
cosbddy
sincos
ab
dxdy
(iii)
22222
2
2
2
then ,1
ellipsetheotangent taisif that Show
kbmaby
ax
kmxy
cosax
sinaddx
sinby
cosbddy
sincos
ab
dxdy
mdxdykmxy hen tangent ta is If
(iii)
22222
2
2
2
then ,1
ellipsetheotangent taisif that Show
kbmaby
ax
kmxy
cosax
sinaddx
sinby
cosbddy
sincos
ab
dxdy
mdxdykmxy hen tangent ta is If
sincos i.e.
abm
(iii)
22222
2
2
2
then ,1
ellipsetheotangent taisif that Show
kbmaby
ax
kmxy
cosax
sinaddx
sinby
cosbddy
sincos
ab
dxdy
mdxdykmxy hen tangent ta is If
sincos i.e.
abm
)1(0cossincossin
bam
bam
(iii)
22222
2
2
2
then ,1
ellipsetheotangent taisif that Show
kbmaby
ax
kmxy
cosax
sinaddx
sinby
cosbddy
sincos
ab
dxdy
mdxdykmxy hen tangent ta is If
sincos i.e.
abm
)1(0cossincossin
bam
bam
then;sin,cosat ellipsemeets tangent If bakamb cossin
(iii)
22222
2
2
2
then ,1
ellipsetheotangent taisif that Show
kbmaby
ax
kmxy
cosax
sinaddx
sinby
cosbddy
sincos
ab
dxdy
mdxdykmxy hen tangent ta is If
sincos i.e.
abm
)1(0cossincossin
bam
bam
then;sin,cosat ellipsemeets tangent If bakamb cossin)2(sincos kbam
0coscossin2sin:1 222222 babmma
2222222 sincossin2cos:2 kbabmma
(+)
0coscossin2sin:1 222222 babmma
2222222 sincossin2cos:2 kbabmma
(+)
22222222 cossincossin kbma
0coscossin2sin:1 222222 babmma
2222222 sincossin2cos:2 kbabmma
(+)
22222222 cossincossin kbma 2222 kbma
Exercise 6C; 1, 3, 11, 15, 18a
Komaroni Unit 1