圆锥曲线 / conic section の 推导 / proof

这是不同圆锥曲线的基本性质和推导，包含椭圆、双曲线、和抛物线。

圆锥曲线虽然记起来有点奇怪，但自己推一遍就比较好记了。不算太复杂，而且化简起来很奇妙。

毕竟，它们本身长得就很特殊。

椭圆 / eclipse

- equation / 公式：$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\ /\ \frac{y^2}{a^2}+\frac{x^2}{b^2}=1\quad\left(a>b>0\right)$
- vertices / 顶点：$(\pm a,0),(0,\pm b)\ /\ (0,\pm a),(\pm b,0)$
- focal points / foci / 焦点：$(\pm c,0)\ /\ (0,\pm c)$ or $(\sqrt{a^2-b^2},0)\ /\ (0,\sqrt{a^2-b^2})$
- directrix / 准线：$x=\pm\frac{a^2}{c}=\pm\frac{a}{e}\ /\ y=\pm\frac{a^2}{c}$
- eccentricity / 离心率：$e=\frac{c}{a}=\frac{\sqrt{a^2-b^2}}{a}<1$
- 面积：$S=\pi ab$

定义：椭圆上任意点到2个焦点距离之和为$2a$

推导1：

$\begin{align}
\sqrt{(x-c)^{2}+y^{2}}+\sqrt{(x+c)^{2}+y^{2}}&=2a\\
\sqrt{(x+c)^{2}+y^{2}}&=2a-\sqrt{(x-c)^{2}+y^{2}}\\
(x+c)^{2}+y^{2}&=4a^2-4a\sqrt{(x-c)^{2}+y^{2}}+(x-c)^{2}+y^{2}\\
4cx&=4a^2-4a\sqrt{(x-c)^{2}+y^{2}}\\
cx&=a^2-a\sqrt{(x-c)^{2}+y^{2}}\\
a\sqrt{(x-c)^{2}+y^{2}}&=a^{2}-cx\\
a^2\left(x^2-2cx+c^2+y^2\right)&=a^4-2a^2cx+c^2x^2\\
a^2x^2-2a^2cx+a^2c^2+a^2y^2&=a^4-2a^2cx+c^2x^2\\
\left(a^{2}-c^{2}\right)x^{2}+a^{2}y^{2}&=a^4-a^2c^2=a^{2}\left(a^{2}-c^{2}\right)\\
\frac{x^2}{a^2}+\frac{y^2}{a^{2}-c^{2}}&=1
\end{align}$

$b^2=a^2-c^2,a^2=b^2+c^2,c^2=a^2-b^2,c=\sqrt{a^2-b^2}$

推导2：

$\begin{aligned}
&\sqrt{(x-c)^{2}+y^{2}}+\sqrt{(x+c)^{2}+y^{2}}=2a\\
&4cx=\left((x+c)^{2}+y^{2}\right)-\left((x-c)^{2}+y^{2}\right)\\
=&\left(\sqrt{(x+c)^{2}+y^{2}}+\sqrt{(x-c)^{2}+y^{2}}\right)\cdot\left(\sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}\right)\\
&\sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}=\frac{4cx}{2a}=\frac{2cx}{a}=2ex\\
\Rightarrow&\begin{cases}
\sqrt{(x+c)^{2}+y^{2}}=a+ex=e(\frac{a}{e}+x)=e(\frac{a^2}{c}+x)\\
\sqrt{(x-c)^{2}+y^{2}}=a-ex=e(\frac{a}{e}-x)=e(\frac{a^2}{c}-x)
\end{cases}
\end{aligned}$

双曲线 / hyperbola

- equation / 公式：$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\ /\ \frac{y^2}{a^2}-\frac{x^2}{b^2}=1\quad\left(a>0,b>0\right)$
- vertices / 顶点：$(\pm a,0)\ /\ (0,\pm a)$
- focal points / foci / 焦点：$(\pm c,0)\ /\ (0,\pm c)$ or $(\pm\sqrt{a^2+b^2},0)\ /\ (0,\pm\sqrt{a^2+b^2})$
- directrix / 准线：$x=\pm\frac{a^2}{c}\ /\ y=\pm\frac{a^2}{c}$
- asymptotes / 渐进线：$y=\pm \frac{b}{a} x\ /\ y=\pm \frac{a}{b} x$
- eccentricity / 离心率：$e=\frac{c}{a}=\frac{\sqrt{a^2+b^2}}{a}>1$

推导1：

If $x\ge a$,

$\begin{align}
\sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}&=2a\\
\sqrt{(x+c)^{2}+y^{2}}&=2a+\sqrt{(x-c)^{2}+y^{2}}\\
(x+c)^{2}+y^{2}&=4a^2+4a\sqrt{(x-c)^{2}+y^{2}}+(x-c)^{2}+y^{2}\\
4cx&=4a^2+4a\sqrt{(x-c)^{2}+y^{2}}\\
cx&=a^2+a\sqrt{(x-c)^{2}+y^{2}}\\
cx-a^2&=a\sqrt{(x-c)^{2}+y^{2}}\\
c^2x^2-2a^2cx+a^4&=a^2\left(x^2-2cx+c^2+y^2\right)\\
c^2x^2-2a^2cx+a^4&=a^2x^2-2a^2cx+a^2c^2+a^2y^2\\
\left(c^{2}-a^{2}\right)x^{2}-a^{2}y^{2}&=a^2c^2-a^4=a^{2}\left(c^{2}-a^{2}\right)\\
\frac{x^2}{a^2}-\frac{y^2}{c^{2}-a^{2}}&=1
\end{align}$

If $x\le-a$,

$\begin{align}
\sqrt{(x-c)^{2}+y^{2}}-\sqrt{(x+c)^{2}+y^{2}}&=2a\\
\sqrt{(x-c)^{2}+y^{2}}-2a&=\sqrt{(x+c)^{2}+y^{2}}\\
(x-c)^{2}+y^{2}-4a\sqrt{(x-c)^{2}+y^{2}}+4a^2&=(x+c)^{2}+y^{2}\\
4a^2&=4cx+4a\sqrt{(x-c)^{2}+y^{2}}\\
a^2&=cx+a\sqrt{(x-c)^{2}+y^{2}}\\
a^2-cx&=a\sqrt{(x-c)^{2}+y^{2}}\\
c^2x^2-2a^2cx+a^4&=a^2\left(x^2-2cx+c^2+y^2\right)\\
c^2x^2-2a^2cx+a^4&=a^2x^2-2a^2cx+a^2c^2+a^2y^2\\
\left(c^{2}-a^{2}\right)x^{2}-a^{2}y^{2}&=a^2c^2-a^4=a^{2}\left(c^{2}-a^{2}\right)\\
\frac{x^2}{a^2}-\frac{y^2}{c^{2}-a^{2}}&=1
\end{align}$

$b^2=c^2-a^2,a^2=c^2-b^2,c^2=a^2+b^2,c=\sqrt{a^2+b^2}$

推导2：

$\begin{aligned}
&\sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}=\begin{cases}2a&x\ge a\\-2a&x\le-a\end{cases}\\
&4cx=\left((x+c)^{2}+y^{2}\right)-\left((x-c)^{2}+y^{2}\right)\\
=&\left(\sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}\right)\cdot\left(\sqrt{(x+c)^{2}+y^{2}}+\sqrt{(x-c)^{2}+y^{2}}\right)\\
&\sqrt{(x+c)^{2}+y^{2}}+\sqrt{(x-c)^{2}+y^{2}}=\begin{cases}\frac{4cx}{2a}=\frac{2cx}{a}=2ex&x\ge a\\\frac{4cx}{-2a}=\frac{-2cx}{a}=-2ex&x\le-a\end{cases}\\
\Rightarrow&\begin{cases}
\sqrt{(x+c)^{2}+y^{2}}=\begin{cases}a+ex&x\ge a\\-a-ex&x\le-a\end{cases}=\left|a+ex\right|=e\left|\frac{a}{e}+x\right|=e\left|\frac{a^2}{c}+x\right|\\
\sqrt{(x-c)^{2}+y^{2}}=\begin{cases}ex-a&x\ge a\\a-ex&x\le-a\end{cases}=\left|ex-a\right|=e\left|\frac{a}{e}-x\right|=e\left|\frac{a^2}{c}-x\right|
\end{cases}
\end{aligned}$

抛物线 / parabola

- equation / 公式：$y^2=2px\ /\ y^2=-2px\ /\ x^2=2py\ /\ x^2=-2py\quad\left(p>0\right)$
- vertex / 顶点：$(0,0)$
- focal points / foci / 焦点：$(\frac{p}{2},0)\ /\ (-\frac{p}{2},0)\ /\ (0,\frac{p}{2})\ /\ (0,-\frac{p}{2})$
- directrix / 准线：$x=-\frac{p}{2}\ /\ x=\frac{p}{2}\ /\ y=-\frac{p}{2}\ /\ y=\frac{p}{2}$
- eccentricity / 离心率：$e=1$

$\begin{array}{c|c}
&\text{vertex}&\text{directrix}&\text{equation}&\text{other points}\\\hline
1:\quad|(&(k,0)&x=-k&(x-k)^2+y^2=(x+k)^2\Longrightarrow y^2=4kx&(k,2k),(k,-2k)\\\hline
2:\quad)|&(-k,0)&x=k&(x+k)^2+y^2=(x-k)^2\Longrightarrow y^2=-4kx\\\hline
3:\quad◡&(0,k)&y=-k&x^2+(y-k)^2=(y+k)^2\Longrightarrow x^2=4ky\\\hline
4:\quad◠&(0,-k)&y=k&x^2+(y+k)^2=(y-k)^2\Longrightarrow x^2=-4ky
\end{array}$