解析几何

椭圆

第一定义

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1(c^2 = a^2 - {b^2} )$ $e =\frac ca= \sqrt {1 - \frac{b^2}{a^2} } \in (0,1)$

第二定义

$准线\\\left\{ \begin{gathered} {l_1} = \frac{a^2}{c} > a \hfill \\ {l_2} = - \frac{a^2}{c} < - a \hfill \\ \end{gathered} \right.$ $\left\{ \begin{gathered} \left| {P{F_1} } \right| = a - e \cdot {x_P} \hfill \\ \left| {P{F_2} } \right| = a + e \cdot {x_P} \hfill \\ \end{gathered} \right.$ $通径H=\frac{2b^2}{a^2}$

$S_{\Delta P{F_1}{F_2} } = {b^2} \cdot \tan \frac{\ang F_1PF_2}{2}$ $\Delta P{F_1}{F_2}的外切圆B切x轴于顶点A$

第三定义

$k_{PF_1} \cdot k_{PF_2} = - \frac{b^2}{a^2}$

$中点弦\\ k_{OM} \cdot k_{PB}=k_{PA} \cdot k_{PB} = - \frac{b^2}{a^2}$

$e\cos \theta = \left| \frac{1 - \lambda }{1 + \lambda } \right|,(\lambda = \frac{AF_1}{BF_1})$

极坐标

$\rho = \frac{ep}{1 - e\cos \theta },(焦准距p = \frac{b^2}{c})$ $L = \left| \frac{H}{ {1 - {e^2}\cos ^2}\theta } \right|,(H=\frac{2b^2}{a}=2ep)$

参数方程

$\left\{ \begin{gathered} x = a\cos t \hfill \\ y = b\sin t \hfill \\ \end{gathered} \right.$

双曲线

第一定义

$\frac{ { {x^2} } } { { {a^2 } } }- \frac{ { {y^2} } }{ { {b^2} } } = 1(c = \sqrt { {a^2} + {b^2 } } )$ $\left| \left| {P{F_2} } \right| - \left| {P{F_1} } \right| \right| = 2a \Leftrightarrow \left| {P{F_2} } \right| - \left| {P{F_1} } \right| = \pm 2a$

第二定义

$e = \frac{c}{a} = \sqrt {1 + \frac{b^2}{a^2} } > 1$ $\left\{ \begin{gathered} \left| {P{F_1} } \right| = e{x_P} - a \hfill \\ \left| {P{F_2} } \right| = e{x_P} + a \hfill \\ \end{gathered} \right.$

性质

$渐近线\\ k = \pm \frac{b}{a},O{E_1} \bot {E_1}{F_1}$ $O{E_1} = a,{E_1}{F_1} = b$

${b^2} = (c - a)(c + a) = {c^2} - {a^2}$

$\Delta P{F_1}{F_2}的内切圆M切x轴于顶点A$ $S_\Delta P{F_1}{F_2} = \frac{ {b^2} }{ \tan \frac{ \angle {F_1}P{F_2} }{2} }$

$|P\overline P | = |Q\overline Q |,{k_{OM} } \cdot {k_{PQ } } = \frac{ { {b^2} } }{ { {a^2} } }\\\left\{ {\begin{array}{*{20}{l} } {x = \frac{a}{ {\cos t} } = a\sec t,} \\ {y = \pm b\tan t} \end{array} } \right.$

抛物线

$\left\{ {\begin{array}{*{20}{l} } { {y^2} = 2px,(x为对称轴)} \\ { {x^2} = 2py,(y为对称轴)} \end{array} } \right.$ $|PF|=|Pl|$

焦点弦

$\left\{ \begin{gathered} \left| {FP} \right| = \frac{p}{ {1 - \cos \theta } } \hfill \\ \left| {FQ} \right| = \frac{p}{ {1 + \cos \theta } } \hfill \\ \end{gathered} \right. \Rightarrow \left| {PQ} \right| = \frac{ {2p} }{ { { {\sin }^2}\theta } }$ ${S_{\Delta PFQ} } = \frac{ { {p^2} } }{ {2\sin \theta } }$ $\left\{ \begin{gathered} {x_p} \cdot {x_Q} = \frac{ { {p^2} } }{4} \hfill \\ {y_P} \cdot {y_Q} = - {p^2} \hfill \\ \end{gathered} \right.$ $\frac{1}{ {\left| {FP} \right|} } + \frac{1}{ {\left| {FQ} \right|} } = \frac{2}{p}$ ${k_{PQ} } \cdot {y_M} = p$

$\begin{gathered} MN = NP = NQ \hfill \\ AM = AN \hfill \\ \left\{ \begin{gathered} \left| {PF} \right| = \left| {AP} \right| = {x_P} + \frac{p}{2} \hfill \\ \left| {QF} \right| = \left| {BQ} \right| = {x_Q} + \frac{p}{2} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered}$ $MP,MQ为切线$

参数方程

$\left\{ \begin{gathered} x = 2p{t^2} \hfill \\ y = 2pt \hfill \\ \end{gathered} \right.$

切线

$\frac {x^2} {a^2}\pm \frac {y^2} {b^2}=1\\ \frac{ {2{x_0} } }{ { {a^2} } } + \frac{ {2{y_0} } }{ { {b^2} } } \cdot \frac{ { {\text{d} }y} }{ { {\text{d} }x} } = 0 \Rightarrow \frac{ { {\text{d} }y} }{ { {\text{d} }x} } = - \frac{ { {b^2}{x_0} } }{ { {a^2}{y_0} } }$

切线方程为

$\frac{ {x_0}x }{ {a^2} } + \frac{ {y_0}y }{ {b^2} } = 1$ $y=kx\pm\sqrt{a^2k^2+b^2}$

抛物线

$\begin{gathered} {y^2} = 2px \hfill \\ 2{y_0} \cdot \frac{ { {\text{d} }y} }{ { {\text{d} }x} } = 2p \Rightarrow \frac{ { {\text{d} }y} }{ { {\text{d} }x} } = \frac{p}{ { {y_0} } } \hfill \\ \end{gathered}$

双曲线

$\frac{ {x_0}x }{ {a^2} } - \frac{ {y_0}y }{ {b^2} } = 1$ $k=\frac{ { {b^2}{x_0} } }{ { {a^2}{y_0} } }\Rightarrow x_0=\frac{ak}{\sqrt{k^2-1}}$ $y=kx\pm\sqrt{a^2k^2-b^2}$ $\frac{ {x_0}x }{ {a^2} } \pm \frac{ {y_0}y }{ {b^2} } = 1\\ y=kx\pm\sqrt{a^2k^2\pm b^2}$

椭圆切线

$y=kx\pm\sqrt{a^2k^2+ b^2}$

双曲线切线

$y=kx\pm\sqrt{a^2k^2- b^2}\textcolor{red} {(|k| > \frac{b}{a} )}$

不包括斜率不存在时

$P({x_0},{y_0})$在切线上，解出$k$

${y_0} - k{x_0} = \pm \sqrt {a^2k^2 - b^2}\\ ({x_0}^2 - {a^2}){k^2} - 2{x_0}{y_0}k + ({y_0}^2 - {b^2}) = 0$

若$k^2$前系数为$0$，则为一次方程，有一条斜率存在的切线，还有一条不存在的切线

若$k^2$前的系数不为$0$，则该方程是一个关于$k$的一元二次方程，可能有$0$个，$1$个，$2$个解

$\begin{array}{l} \Delta &= 4x_0y_0 - 4(x_0^2 - a^2)(y_0^2 + b^2)\\ &=4{a^2}{b^2}\left[\textcolor{red}{ 1 - \left(\frac{x_0^2}{a^2} - \frac{y_0^2}{b^2} \right)}\right] \end{array}$

若$\frac{x_0^2}{a^2} - \frac{y_0^2}{b^2}=0$，有一根为$k=\frac{b}{a}/ - \frac{b}{a}$，故舍去