双曲線

双曲線の定義 (焦点がx軸上の場合)
双曲線の方程式 (焦点がx軸上の場合)
双曲線の方程式の漸近線 (焦点がx軸上の場合)
双曲線の定義 (焦点がy軸上の場合)
双曲線の方程式 (焦点がy軸上の場合)
双曲線の方程式の漸近線 (焦点がy軸上の場合)

■双曲線の定義(焦点がx軸上の場合)

平面上で， $3$\displaystyle{2}$ 定点 $3$\displaystyle{\hspace{1}{F}_{1}}$ ， $3$\displaystyle{\hspace{1}{F}_{2}}$ からの距離の差が一定で点の軌跡を双曲線といい

点 $3$\displaystyle{\hspace{1}{F}_{1}}$ ， $3$\displaystyle{\hspace{1}{F}_{2}}$ を焦点という．

$Hyperbola 1$
下図の点Pを動かして定義を確かめてみよう

JSXGraph Copyright (C) see http://jsxgraph.org

$3$\displaystyle{c> 0}$ のとき，焦点座標は $3$\displaystyle{\hspace{1}{F}_{1} $ c,0 $ }$ ， $3$\displaystyle{\hspace{1}{F}_{2} $ - c,0 $ }$ である．

$3$\displaystyle{P $ x,y $ }$ であるから

$3$\displaystyle{\hspace{1}{F}_{1}P=\sqrt{ { \| x- c \| }^{2}+{ \|y\| }^{2} }}$ ， $3$\displaystyle{\hspace{1}{F}_{2}P=\sqrt{ { \| x- $ - c $ \| }^{2}+{ \|y\| }^{2} }}$

$3$\displaystyle{ \| \hspace{1}{F}_{1}P- \hspace{1}{F}_{2}P \| =2a}$ $3$\displaystyle{ $ c> a $ }$ となる．

$3$\displaystyle{2a}$ は双曲線の $3$\displaystyle{x}$ 軸との交点の距離の差である．

$3$\displaystyle{2a=a- $- a$}$

より，双曲線の $3$\displaystyle{x}$ 軸との交点の $3$\displaystyle{x}$ 座標の値は $3$\displaystyle{a}$ ， $3$\displaystyle{- a}$ である．

この双曲線を表す方程式は

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$ 　(ただし， $3$\displaystyle{ b=\sqrt{ {c}^{2}- {a}^{2} } }$ )

となる．

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■双曲線の方程式 $3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$ の導出(焦点がx軸上の場合)

◆ $3$\displaystyle{\hspace{1}{F}_{2}P> \hspace{1}{F}_{1}P}$ の時

$3$\displaystyle{\sqrt{ { \| x- $ - c $ \| }^{2}+{ \|y\| }^{2} }- \sqrt{ { \| x- c \| }^{2}+{ \|y\| }^{2} }=2a}$

$3$\displaystyle{\sqrt{ { $ x+c $ }^{2}+{y}^{2} }=2a+\sqrt{ { $ x- c $ }^{2}+{y}^{2} }}$

両辺を $3$\displaystyle{2}$ 乗して，整理すると

$3$\displaystyle{{ $ x+c $ }^{2}+{y}^{2}}$ $3$\displaystyle{=2a+4a\sqrt{ { $ x- c $ }^{2}+{y}^{2} }+{ $ x- c $ }^{2}+{y}^{2}}$

$3$\displaystyle{2cx=4{a}^{2}+4a\sqrt{ { $ x- c $ }^{2}+{y}^{2} }- 2cx}$

$3$\displaystyle{4a\sqrt{ { $ x- c $ }^{2}+{y}^{2} }=4cx- 4{a}^{2}}$

$3$\displaystyle{\sqrt{ { $ x- c $ }^{2}+{y}^{2} }=\frac{\hspace{2}c\hspace{2}}{\hspace{2}a\hspace{2}}x- a}$

更に両辺を $3$\displaystyle{2}$ 乗する．

$3$\displaystyle{{ $ x- c $ }^{2}+{y}^{2}={ $ \frac{\hspace{2}c\hspace{2}}{\hspace{2}a\hspace{2}}x- a $ }^{2}}$

$3$\displaystyle{{x}^{2}+{c}^{2}+{y}^{2}=\frac{\hspace{2} {c}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{x}^{2}+{a}^{2}}$

$3$\displaystyle{ $ 1- \frac{\hspace{2} {c}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}} $ {x}^{2}+{y}^{2}={a}^{2}- {c}^{2}}$

$3$\displaystyle{\frac{\hspace{2} {a}^{2}- {c}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{x}^{2}+{y}^{2}={a}^{2}- {c}^{2}}$

両辺を $3$\displaystyle{ {a}^{2}- {c}^{2} }$ で割る．

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}+\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {a}^{2}- {c}^{2} \hspace{2}}=1}$

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {c}^{2}- {a}^{2} \hspace{2}}=1}$

$3$\displaystyle{c> a}$ より $3$\displaystyle{{c}^{2}- {a}^{2}> 0}$ となる．よって $3$\displaystyle{{c}^{2}- {a}^{2}={b}^{2}}$ ， $3$\displaystyle{b> 0}$ とおくと

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$

となる．

◆ $3$\displaystyle{\hspace{1}{F}_{1}P> \hspace{1}{F}_{2}P}$ のとき

$3$\displaystyle{\sqrt{ { \| x- c \| }^{2}+{ \|y\| }^{2} }- \sqrt{ { \| x- $ - c $ \| }^{2}+{ \|y\| }^{2} }=2a}$

$3$\displaystyle{\sqrt{ { $ x- c $ }^{2}+{y}^{2} }=2a- \sqrt{ { $ x+c $ }^{2}+{y}^{2} }}$

両辺を2乗する．

$3$\displaystyle{{ $ x- c $ }^{2}+{y}^{2}}$ $3$\displaystyle{={ $ 2a $ }^{2}+4a\sqrt{ { $ x+c $ }^{2}+{y}^{2} }+{ $ x+c $ }^{2}+{y}^{2}}$

$3$\displaystyle{{x}^{2}- 2cx+{c}^{2}+{y}^{2}}$ $3$\displaystyle{=4{a}^{2}+4a\sqrt{ { $ x+c $ }^{2}+{y}^{2} }+{x}^{2}+2cx+{c}^{2}+{y}^{2}}$

左辺と右辺を整理して

$3$\displaystyle{- 2cx=4{a}^{2}+4a\sqrt{ { $ x+c $ }^{2}+{y}^{2} }+2cx}$

$3$\displaystyle{4a\sqrt{ { $ x+c $ }^{2}+{y}^{2} }=- 4cx- 4{a}^{2}}$

$3$\displaystyle{\sqrt{ { $ x+c $ }^{2}+{y}^{2} }=- \frac{\hspace{2}c\hspace{2}}{\hspace{2}a\hspace{2}}x- a}$

更に両辺を2乗する．

$3$\displaystyle{{ $ x+c $ }^{2}+{y}^{2}={ $ - \frac{\hspace{2}c\hspace{2}}{\hspace{2}a\hspace{2}}x- a $ }^{2}}$

$3$\displaystyle{{x}^{2}+2cx+{c}^{2}+{y}^{2}}$ $3$\displaystyle{=\frac{\hspace{2} {c}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{x}^{2}+2cx+{a}^{2}}$

$3$\displaystyle{{x}^{2}+{c}^{2}+{y}^{2}=\frac{\hspace{2} {c}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{x}^{2}+{a}^{2}}$

$3$\displaystyle{ $ 1- \frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2} {c}^{2} \hspace{2}} $ {x}^{2}+{y}^{2}={a}^{2}- {c}^{2}}$

$3$\displaystyle{\frac{\hspace{2} {a}^{2}- {c}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{x}^{2}+{y}^{2}={a}^{2}- {c}^{2}}$

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}+\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {a}^{2}- {c}^{2} \hspace{2}}=1}$

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}+\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} - $ {c}^{2}- {a}^{2} $ \hspace{2}}=1}$

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {c}^{2}- {a}^{2} \hspace{2}}=1}$

$3$\displaystyle{c> a}$ より $3$\displaystyle{{c}^{2}- {a}^{2}> 0}$ となる．よって $3$\displaystyle{{c}^{2}- {a}^{2}={b}^{2}}$ ， $3$\displaystyle{b> 0}$ とおくと

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$

となる．

このように， $3$\displaystyle{\hspace{1}{F}_{2}P> \hspace{1}{F}_{1}P}$ のときと， $3$\displaystyle{\hspace{1}{F}_{1}P> \hspace{1}{F}_{2}P}$ のときで，双曲線の方程式が等しくなる．

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■双曲線の方程式 $3$\displaystyle{ \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1 }$ の漸近線(焦点がx軸上の場合)

双曲線の方程式 $3$\displaystyle{ \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1 }$ より

$3$\displaystyle{\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- 1}$

$3$\displaystyle{{y}^{2}=\frac{\hspace{2} {b}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{x}^{2}- {b}^{2}}$

$3$\displaystyle{{y}^{2}=\frac{\hspace{2} {b}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{x}^{2} $ 1- \frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} $ }$

$3$\displaystyle{y=\pm \frac{\hspace{2}b\hspace{2}}{\hspace{2}a\hspace{2}}x\sqrt{ 1- \frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} }}$

したがって， $3$\displaystyle{y=\pm \frac{\hspace{2}b\hspace{2}}{\hspace{2}a\hspace{2}}x\sqrt{ 1- 0 }}$ より， $3$\displaystyle{ \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1 }$ の漸近線は

$3$\displaystyle{y=\pm \frac{\hspace{2}b\hspace{2}}{\hspace{2}a\hspace{2}}x}$

となる．

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■双曲線の定義(焦点がy軸上の場合)

$Hperbola 2$

$3$\displaystyle{\hspace{1}{F}_{1} $ 0,c $ }$

$3$\displaystyle{\hspace{1}{F}_{2} $ 0,- c $ }$

$3$\displaystyle{\hspace{1}{F}_{1}P=\sqrt{ { \|x\| }^{2}+{ \| y- c \| }^{2} }}$

$3$\displaystyle{\hspace{1}{F}_{2}P=\sqrt{ { \|x\| }^{2}+{ \| y- $ - c $ \| }^{2} }}$

$3$\displaystyle{ \| \hspace{1}{F}_{1}P- \hspace{1}{F}_{2}P \| =2b}$

$3$\displaystyle{c> b}$

距離の差 $3$\displaystyle{2b}$

双曲線の $3$\displaystyle{y}$ 軸との交点の $3$\displaystyle{y}$ 座標の値は $3$\displaystyle{b,- b}$

この双曲線を表す方程式は

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=- 1}$ 　(ただし， $3$\displaystyle{a=\sqrt{ {c}^{2}- {b}^{2} }}$ )

となる．

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■双曲線の方程式 $3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=- 1}$ の導出(焦点がy軸上の場合)

◆ $3$\displaystyle{\hspace{1}{F}_{2}P> \hspace{1}{F}_{1}P}$ の時

$3$\displaystyle{\sqrt{ { \|x\| }^{2}+{ \| y- $ - c $ \| }^{2} }- \sqrt{ { \|x\| }^{2}+{ \| y- c \| }^{2} }=2b}$

$3$\displaystyle{\sqrt{ {x}^{2}+{ $ y+c $ }^{2} }}$ $3$\displaystyle{=2b+\sqrt{ {x}^{2}+{ $ y- c $ }^{2} }}$

$3$\displaystyle{{x}^{2}+{ $ y+c $ }^{2}}$ $3$\displaystyle{={ $ 2b $ }^{2}+4b\sqrt{ {x}^{2}+{ $ y- c $ }^{2} }+{x}^{2}+{ $ y- c $ }^{2}}$

$3$\displaystyle{{x}^{2}+{y}^{2}+2cy+{c}^{2}}$ $3$\displaystyle{=4{b}^{2}+4b\sqrt{ {x}^{2}+{ $ y- c $ }^{2} }+{x}^{2}+{y}^{2}- 2cy+{c}^{2}}$

$3$\displaystyle{4b\sqrt{ {x}^{2}+{ $ y- c $ }^{2} }=4cy- 4{b}^{2}}$

$3$\displaystyle{\sqrt{ {x}^{2}+{ $ y- c $ }^{2} }=\frac{\hspace{2}c\hspace{2}}{\hspace{2}b\hspace{2}}y- b}$

$3$\displaystyle{{x}^{2}+{ $ y- c $ }^{2}={ $ \frac{\hspace{2}c\hspace{2}}{\hspace{2}b\hspace{2}}y- b $ }^{2}}$

$3$\displaystyle{{x}^{2}+{y}^{2}- 2cy+{c}^{2}=\frac{\hspace{2} {c}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}{y}^{2}- 2cy+{b}^{2}}$

$3$\displaystyle{{x}^{2}+ $ 1- \frac{\hspace{2} {c}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}} $ {y}^{2}={b}^{2}- {c}^{2}}$

$3$\displaystyle{{x}^{2}+\frac{\hspace{2} {b}^{2}- {c}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}{y}^{2}={b}^{2}- {c}^{2}}$

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {b}^{2}- {c}^{2} \hspace{2}}+\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} - $ {c}^{2}- {b}^{2} $ \hspace{2}}+\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$

$3$\displaystyle{c> b}$ より $3$\displaystyle{{c}^{2}- {b}^{2}> 0}$ となる．よって $3$\displaystyle{{c}^{2}- {b}^{2}={a}^{2}}$ ， $3$\displaystyle{a> 0}$ とおくと $3$\displaystyle{{c}^{2}- {b}^{2}={a}^{2}}$ とおくと

$3$\displaystyle{- \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}+\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=- 1}$

となる．

◆ $3$\displaystyle{\hspace{1}{F}_{1}P> \hspace{1}{F}_{2}P}$ の時

$3$\displaystyle{\sqrt{ { \|x\| }^{2}+{ \| y- c \| }^{2} }- \sqrt{ { \|x\| }^{2}+{ \| y- $ - c $ \| }^{2} }=2b}$

$3$\displaystyle{{x}^{2}+{ $ y- c $ }^{2}}$ $3$\displaystyle{={ $ 2b $ }^{2}+4b\sqrt{ {x}^{2}+{ $ y+c $ }^{2} }+{x}^{2}+{ $ y+c $ }^{2}}$

$3$\displaystyle{{x}^{2}+{y}^{2}- 2cy+{c}^{2}}$ $3$\displaystyle{=4{b}^{2}+4b\sqrt{ {x}^{2}+{ $ y+c $ }^{2} }+{x}^{2}+{y}^{2}+2cy+{c}^{2}}$

$3$\displaystyle{4b\sqrt{ {x}^{2}+{ $ y+c $ }^{2} }}$ $3$\displaystyle{=- 4cy- 4{b}^{2}}$

$3$\displaystyle{\sqrt{ {x}^{2}+{ $ y+c $ }^{2} }}$ $3$\displaystyle{=- \frac{\hspace{2}c\hspace{2}}{\hspace{2}b\hspace{2}}y- b}$

$3$\displaystyle{{x}^{2}+{ $ y+c $ }^{2}={ $ - \frac{\hspace{2}c\hspace{2}}{\hspace{2}b\hspace{2}}y- b $ }^{2}}$

$3$\displaystyle{{x}^{2}+{y}^{2}+2cy+{c}^{2}}$ $3$\displaystyle{=\frac{\hspace{2} {c}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}{y}^{2}+2cy+{b}^{2}}$

$3$\displaystyle{{x}^{2}+ $ 1- \frac{\hspace{2} {c}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}} $ {y}^{2}={b}^{2}- {c}^{2}}$

$3$\displaystyle{{x}^{2}+\frac{\hspace{2} {b}^{2}- {c}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}{y}^{2}={b}^{2}- {c}^{2}}$

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {b}^{2}- {c}^{2} \hspace{2}}+\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} - $ {c}^{2}- {b}^{2} $ \hspace{2}}+\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$

$3$\displaystyle{- \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {c}^{2}- {b}^{2} \hspace{2}}+\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$

$3$\displaystyle{- \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}+\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$

両辺に $3$\displaystyle{- 1}$ を掛けて

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=- 1}$

となる．

このように， $3$\displaystyle{\hspace{1}{F}_{2}P> \hspace{1}{F}_{1}P}$ の場合と $3$\displaystyle{\hspace{1}{F}_{1}P> \hspace{1}{F}_{2}P}$ の場合で方程式が等しくなる．

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■双曲線の方程式 $3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=- 1}$ の漸近線(焦点がy軸上の場合)

$3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=- 1}$ より

$3$\displaystyle{\frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}+1}$

$3$\displaystyle{{y}^{2}=\frac{\hspace{2} {b}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{x}^{2}+{b}^{2}}$

$3$\displaystyle{{y}^{2}=\frac{\hspace{2} {b}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{x}^{2} $ 1+\frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} $ }$

$3$\displaystyle{y=\pm \frac{\hspace{2}b\hspace{2}}{\hspace{2}a\hspace{2}}x\sqrt{ 1+\frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} }}$

$3$\displaystyle{x\rightarrow \infty}$ あるいは， $3$\displaystyle{x\rightarrow - \infty}$ の時， $3$\displaystyle{\frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}}\rightarrow 0}$ となるので， $3$\displaystyle{y}$ の値は $3$\displaystyle{\pm \frac{\hspace{2}b\hspace{2}}{\hspace{2}a\hspace{2}}x}$ に近づく．

すなわち， $3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=- 1}$ のグラフの漸近線は

$3$\displaystyle{y=\pm \frac{\hspace{2}b\hspace{2}}{\hspace{2}a\hspace{2}}x}$

となる．

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最終更新日： 2024年7月24日

双曲線

目次

■双曲線の定義(焦点がx軸上の場合)

■双曲線の方程式 の導出(焦点がx軸上の場合)

◆ の時

◆ のとき

■双曲線の方程式 の漸近線(焦点がx軸上の場合)

■双曲線の定義(焦点がy軸上の場合)

■双曲線の方程式 の導出(焦点がy軸上の場合)

◆ の時

◆ の時

■双曲線の方程式 の漸近線(焦点がy軸上の場合)

■双曲線の方程式 $3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1}$ の導出(焦点がx軸上の場合)

◆ $3$\displaystyle{\hspace{1}{F}_{2}P> \hspace{1}{F}_{1}P}$ の時

◆ $3$\displaystyle{\hspace{1}{F}_{1}P> \hspace{1}{F}_{2}P}$ のとき

■双曲線の方程式 $3$\displaystyle{ \frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=1 }$ の漸近線(焦点がx軸上の場合)

■双曲線の方程式 $3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=- 1}$ の導出(焦点がy軸上の場合)

◆ $3$\displaystyle{\hspace{1}{F}_{2}P> \hspace{1}{F}_{1}P}$ の時

◆ $3$\displaystyle{\hspace{1}{F}_{1}P> \hspace{1}{F}_{2}P}$ の時

■双曲線の方程式 $3$\displaystyle{\frac{\hspace{2} {x}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {b}^{2} \hspace{2}}=- 1}$ の漸近線(焦点がy軸上の場合)