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Elements of the Differential and Integral Calculus - Granville - Revised.djvu/213
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8.
The curve
{
x
=
a
(
cos
t
+
t
sin
t
)
,
y
=
a
(
sin
t
−
t
cos
t
)
.
{\displaystyle {\begin{cases}x=a(\cos t+t\sin t),\\y=a(\sin t-t\cos t).\end{cases}}}
Ans.
{
α
=
a
cos
t
,
β
=
a
sin
t
.
{\displaystyle {\begin{cases}\alpha =a\cos t,\\\beta =a\sin t.\end{cases}}}
9.
The curve
{
x
=
3
t
,
y
=
t
2
−
6.
{\displaystyle {\begin{cases}x=3t,\\y=t^{2}-6.\end{cases}}}
{
α
=
−
4
3
t
3
,
β
=
3
t
2
−
3
2
.
{\displaystyle {\begin{cases}\alpha =-{\frac {4}{3}}t^{3},\\\beta =3t^{2}-{\frac {3}{2}}.\end{cases}}}
10.
The curve
{
x
=
6
−
t
2
y
=
2
t
.
{\displaystyle {\begin{cases}x=6-t^{2}\\y=2t.\end{cases}}}
{
α
=
4
−
3
t
2
,
β
=
−
2
t
3
.
{\displaystyle {\begin{cases}\alpha =4-3t^{2},\beta =-2t^{3}.\end{cases}}}
11.
The curve
{
x
=
2
t
,
y
=
t
2
−
2.
{\displaystyle {\begin{cases}x=2t,\\y=t^{2}-2.\end{cases}}}
{
α
=
−
2
t
3
,
β
=
3
t
2
.
{\displaystyle {\begin{cases}\alpha =-2t^{3},\\\beta =3t^{2}.\end{cases}}}
12.
The curve
{
x
=
4
t
,
y
=
3
+
t
2
.
{\displaystyle {\begin{cases}x=4t,\\y=3+t^{2}.\end{cases}}}
{
α
=
−
t
3
,
β
=
11
+
3
t
2
.
{\displaystyle {\begin{cases}\alpha =-t^{3},\\\beta =11+3t^{2}.\end{cases}}}
13.
The curve
{
x
=
9
−
t
2
,
y
=
2
t
.
{\displaystyle {\begin{cases}x=9-t^{2},\\y=2t.\end{cases}}}
{
α
=
7
−
3
t
2
,
β
=
−
2
t
3
.
{\displaystyle {\begin{cases}\alpha =7-3t^{2},\beta =-2t^{3}.\end{cases}}}
14.
The curve
{
x
=
2
t
,
y
=
1
3
t
3
.
{\displaystyle {\begin{cases}x=2t,\\y={\frac {1}{3}}t^{3}.\end{cases}}}
{
α
=
4
t
−
t
5
4
.
β
=
12
+
5
t
4
6
t
.
{\displaystyle {\begin{cases}\alpha ={\frac {4t-t^{5}}{4}}.\\\beta ={\frac {12+5t^{4}}{6t}}.\end{cases}}}
15.
The curve
{
x
=
1
3
t
3
,
y
=
t
2
.
{\displaystyle {\begin{cases}x={\frac {1}{3}}t^{3},\\y=t^{2}.\end{cases}}}
{
α
=
4
t
3
+
12
t
3
β
=
−
2
t
2
+
t
4
2
.
{\displaystyle {\begin{cases}\alpha ={\frac {4t^{3}+12t}{3}}\\\beta =-{\frac {2t^{2}+t^{4}}{2}}.\end{cases}}}
16.
The curve
{
x
=
2
t
,
y
=
3
t
.
{\displaystyle {\begin{cases}x=2t,\\y={\frac {3}{t}}.\end{cases}}}
{
α
=
12
t
4
+
9
4
t
3
β
=
27
+
4
t
4
6
t
.
{\displaystyle {\begin{cases}\alpha ={\frac {12t^{4}+9}{4t^{3}}}\\\beta ={\frac {27+4t^{4}}{6t}}.\end{cases}}}
17.
x
=
4
−
t
2
,
y
=
2
t
.
{\displaystyle x=4-t^{2},\ y=2t.}
22.
x
=
t
,
y
=
t
3
.
{\displaystyle x=t,\ y=t^{3}.}
18.
x
=
2
t
,
y
=
16
−
t
2
.
{\displaystyle x=2t,\ y=16-t^{2}.}
23.
x
=
sin
t
,
y
=
3
cos
t
.
{\displaystyle x=\sin t,\ y=3\cos t.}
19.
x
=
t
,
y
=
sin
t
.
{\displaystyle x=t,\ y=\sin t.}
24.
x
=
1
−
cos
t
,
y
=
t
−
sin
t
.
{\displaystyle x=1-\cos t,\ y=t-\sin t.}
20.
x
=
4
t
,
y
=
3
t
.
{\displaystyle x={\frac {4}{t}},\ y=3t.}
25.
x
=
cos
4
t
,
y
=
sin
4
t
.
{\displaystyle x=\cos ^{4}t,\ y=\sin ^{4}t.}
21.
x
=
t
2
,
y
=
1
6
t
3
.
{\displaystyle x=t^{2},\ y={\frac {1}{6}}t^{3}.}
26.
x
=
a
sec
t
,
y
=
b
tan
t
.
{\displaystyle x=a\sec t,\ y=b\tan t.}