圆锥曲线基本性质

圆

以$A(x_1,y_1)$,$B(x_2,y_2)$为直径两端点的圆

$$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$$
过点$P(x_0,y_0)$的圆$x^2 + y^2 = r^2$的切线与圆交于A、B两点，则 $$l_{AB} : x_0x + y_0y = r^2$$

椭圆

通径 $\frac {2b^2}a$
焦点三角形 $S = b^2\tan \frac {\theta}2$
准线 $x = \frac {a^2}c$
焦半径 $PF_1 = a + ex_0$，$PF_2 = a - ex_0$
椭圆上一点到长轴两端点斜率之积为定值 $e^2 - 1$
过其上一点$P(x_0,y_0)$的切线 $\frac {x_0x}{a^2} + \frac {y_0y}{b^2} = 1$
过其外一点$P(x_0,y_0)$的切点弦 $\frac {x_0x}{a^2} + \frac {y_0y}{b^2} = 1$
中点弦 $k_{AB}k_{OM} = -\frac {b^2}{a^2}$
直径AB $k_{PA}k_{PB} = -\frac {b^2}{a^2}$

双曲线

通径 $\frac {2b^2}a$
焦点三角形 $S = \frac {b^2}{\tan \frac {\theta}2}$
准线 $x = \frac {a^2}c$
焦半径 $PF_1 = |a + ex_0|$，$PF_2 = |a - ex_0|$
过其上一点$P(x_0,y_0)$的切线 $\frac {x_0x}{a^2} - \frac {y_0y}{b^2} = 1$
中点弦 $k_{AB}k_{OM} = \frac {b^2}{a^2}$
直径AB $k_{PA}k_{PB} = \frac {b^2}{a^2}$
焦点到渐进线的距离 = b
焦点三角形内切圆与实轴切于$(\pm a,0)$

抛物线

//$y^2 = 2px$
焦点弦AB :
$y_1y_2 = - p^2$，$x_1x_2 = \frac {p^2}4$
$|AB| = x_1 + x_2 + p = \frac {2p}{\sin^2 \theta}$
$\frac 1{|AF|} + \frac 1{|BF|} = \frac 2p$
$k_{OA}k_{OB} = -4$
$OA \bot OB \iff AB$过(2p,0)
以AB为直径的圆与准线相切
过A、B的切线垂直且交于准线
焦半径 $PF = x_0 + \frac p2$
过其上一点$P(x_0,y_0)$的切线 $y_0y = p(x + x_0)$
过其上两点$P_1(x_1,y_1)$，$P_2(x_2,y_2)$的切线交于点 $M(\frac {y_1y_2}{2p},\frac {y_1 + y_2}2)$