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(1) cos sin sin cos 2 t t θ θ θ θ

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(1)

[  東京工業大学  1994 年前期  2  ] 

    双曲線 xy= −2 Cとする。C上の点 2

, ( 0)

P t t t

を,原点を中心とし反時計回りに角 θだけ回転した点をQとする。

(1) Qの座標をθtで表せ。

(2) θを固定しPC上を動くとき,Qはどのような曲線をえがくか。その方程式を求めよ。

(3) Qのえがく曲線が,点( 3 1,+ 3 1) を通るようなθの値を0< <θ 2π の範囲ですべて求めよ。

  (1) cos sin sin cos 2

t t

θ θ

θ θ

⎞⎜

⎟⎜

⎠⎜

cos 2 sin

sin 2cos

t t

t t

θ θ

θ θ

+

=

より Q tcos 2 sin , sint 2cos

t t

θ θ θ θ

+

(2) Q( ,X Y) とおく。

   

cos 2 sin

sin 2cos X t

t Y t

t

θ θ

θ θ

⎧ = +

⎪⎪

⎪ =

⎪⎩

"

"

であり,

    ①×sinθ−②×cosθ 2 sin cos

X Y

θ θ = t …③

①×cosθ +②×sinθXcosθ +Ysinθ =t …④

③,④より t を消去して,Qのえがく曲線は (XsinθYcosθ)(Xcosθ +Ysinθ)=2

(3) Qのえがく曲線が点( 3 1,+ 3 1) を通るので

( ) ( )

{ 3 1 sin+ θ 3 1 cos θ}{( 3 1 cos+ ) θ +( 3 1 sin ) θ}=2

( 3 1)( 3 1 cos) 2θ ( 3 1)( 3 1 sin) 2θ

+ + + +{( 3 1+ ) (2 3 1 )2}cos sinθ θ =2

2 2

2 cos θ 2sin θ 4 3 sin cosθ θ 2

+ + =

2 2

cos θ 1 cos θ 2 3 sin cosθ θ 1

+ − + =

(2)

( )

cosθ cosθ 3 sinθ =0

よって cosθ =0 または 1 tanθ = 3 0< <θ 2π より 7 3

, , ,

6 2 6 2

π π π π

θ =

参照

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