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(1)

次の点P

( , x y )

は，どのような曲線を描くか求めよ。

(1)

x = 2 cos , θ y = 2 sin θ

2 2

cos θ + sin θ = 1

に

cos , sin

2 2

x y

θ = θ =

を代入して

2 2

2 2 1

x y

  +   =

   

   

⇔

2 2

4 x + y =

よって，原点中心，半径

2

の円。

(2)

x = 2 sin θ + cos , θ y = sin θ − 2 cos θ 2 sin cos

x = θ + θ

…①

sin 2 cos

y = θ − θ

…②

①－②×

2

より

x − 2 y = 4 cos θ

⇔

1

cos ( 2 )

4 x y

θ = −

…③

①×

2

＋②より

2 x + = y 4 sin θ

⇔

1

sin (2 )

4 x y

θ = +

…④

2 2

cos θ + sin θ = 1

に③，④を代入して

2 2

1 1

( 2 ) (2 ) 1

4 x y 4 x y

   

− + + =

   

   

⇔

2 2

5 x + 5 y = 16

⇔ 2 2

16 x + y = 5

よって，原点中心，半径

4 5

5

の円。

(3) 2 2

1 ,

1 1

x y t

t t

= =

+ +

2

1 x 1

= t

+

…①，

1

2

y t

= t

+

…② ①，②より

y = tx

…③

①より

x ≠ 0

であり，

③ ⇔

y

t = x

を①に代入して １３３．媒介変数表示②

(1) 原点中心，半径

2

の円 (2) 原点中心，半径

4 5

5

の円

(3)

1 2 , 0

 

 

 

中心，半径

1

2

の円 (4) 双曲線

x

2

− y

2

= 1

(2)

2

1 1 x

y x

= +      

⇔

2

2 2

x x

x y

= +

⇔

1

2 2

x

x y

= +

⇔

2 2

0 x − + x y =

⇔

2

1

2

1

2 4

x y

 

− + =

 

 

よって，

1 2 , 0

 

 

 

中心，半径

1

2

の円。

(4)

2

2 2

1 2

1 , 1

t t

x y

t t

= + =

− −

2 2

1 1 x t

t

= +

−

…①， 2

2 1 y t

= t

−

…②

①＋②より

2 2

2

1 2 (1 ) 1

1 (1 )(1 ) 1

t t t t

x y

t t t t

+ + + +

+ = = =

− + − −

…③

①－②より

2 2

2

1 2 (1 ) 1

1 (1 )(1 ) 1

t t t t

x y

t t t t

+ − − −

− = = =

− + − +

…④

③，④より

1

x y

x y + = −

⇔

2 2

1 x − y =

よって，双曲線

x

2

− y

2

= 1

参照

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