##### 椭圆 / eclipse
- equation / 公式：$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\ /\ \frac{y^2}{a^2}+\frac{x^2}{b^2}=1\quad\left(a>b>0\right)$
- 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 $(\sqrt{a^2-b^2},0)\ /\ (0,\sqrt{a^2-b^2})$
- The left focus / focal point is $F_1$ and the right focus / focal point is $F_2$.
- 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：$S=\pi ab$
第一定义：
$\begin{align}
\sqrt{(x-c)^{2}+y^{2}}+\sqrt{(x+c)^{2}+y^{2}}&=2a\\
\sqrt{(x+c)^{2}+y^{2}}&=2a-\sqrt{(x-c)^{2}+y^{2}}\\
(x+c)^{2}+y^{2}&=4a^2-4a\sqrt{(x-c)^{2}+y^{2}}+(x-c)^{2}+y^{2}\\
4cx&=4a^2-4a\sqrt{(x-c)^{2}+y^{2}}\\
cx&=a^2-a\sqrt{(x-c)^{2}+y^{2}}\\
a\sqrt{(x-c)^{2}+y^{2}}&=a^{2}-cx\\
a^2\left(x^2-2cx+c^2+y^2\right)&=a^4-2a^2cx+c^2x^2\\
a^2x^2-2a^2cx+a^2c^2+a^2y^2&=a^4-2a^2cx+c^2x^2\\
\left(a^{2}-c^{2}\right)x^{2}+a^{2}y^{2}&=a^4-a^2c^2=a^{2}\left(a^{2}-c^{2}\right)\\
\frac{x^2}{a^2}+\frac{y^2}{a^{2}-c^{2}}&=1
\end{align}$
$b^2=a^2-c^2,a^2=b^2+c^2,c^2=a^2-b^2,c=\sqrt{a^2-b^2}$
第二定义：
\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}
##### 双曲线 / hyperbola
- equation / 公式：$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\ /\ \frac{y^2}{a^2}-\frac{x^2}{b^2}=1\quad\left(a>0,b>0\right)$
- 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：$x=\pm\frac{a^2}{c}\ /\ y=\pm\frac{a^2}{c}$
- 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：
$\begin{align}
&\sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}=\begin{cases}2a&x\ge a\\-2a&x\le-a\end{cases}\\
&4cx=\left((x+c)^{2}+y^{2}\right)-\left((x-c)^{2}+y^{2}\right)=\left(\sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}\right)\cdot\left(\sqrt{(x+c)^{2}+y^{2}}+\sqrt{(x-c)^{2}+y^{2}}\right)\\
&\sqrt{(x+c)^{2}+y^{2}}+\sqrt{(x-c)^{2}+y^{2}}=\begin{cases}\frac{4cx}{2a}&x\ge a\\\frac{4cx}{-2a}&x\le-a\end{cases}=\begin{cases}\frac{2cx}{a}&x\ge a\\\frac{-2cx}{a}&x\le-a\end{cases}=\begin{cases}2ex&x\ge a\\-2ex&x\le-a\end{cases}\\
&\begin{cases}
\sqrt{(x+c)^{2}+y^{2}}=\begin{cases}a+ex&x\ge a\\-a-ex&x\le-a\end{cases}=\left|a+ex\right|=e\left|\frac{a}{e}+x\right|=e\left|\frac{a^2}{c}+x\right|\\
\sqrt{(x-c)^{2}+y^{2}}=\begin{cases}ex-a&x\ge a\\a-ex&x\le-a\end{cases}=\left|ex-a\right|=\left|a+ex\right|=e\left|\frac{a}{e}-x\right|=e\left|\frac{a^2}{c}-x\right|
\end{cases}
\end{align}$
##### 抛物线 / parabola
- equation / 公式：$y^2=2px\ /\ y^2=-2px\ /\ x^2=2py\ /\ x^2=-2py\quad\left(p>0\right)$
- vertice / 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：$e=1$