圆锥曲线椭圆双曲线抛物线的定义和几何性质_双曲线的所有公式[通俗易懂]

1、二次曲线公式

• 二次曲线的代数方程

$$a{x}^2+bxy+c{y}^2+dx+ey+f=0,a、b、c、d、e、f\in R$$

• 二次曲线的齐次方程

$$A{x}^2+2Bxy+C{y}^2+2Dx+2Ey+F=0$$

2、表达式

• 齐次方程的判别式：

$$\Delta =B^2-AC$$

• 代数方程的判别式：

$$\Delta =b^2-4ac$$

• 其他表达式：

$$\begin{gathered} {\delta }_1=f\Delta +(a{e}^2+c{d}^2-bde) \\ {\delta }_2=d^2+e^2-4(a+c)f \\ {\sigma }_1=a{e}^2+c{d}^2-bde \\ {\sigma }_2=\sqrt{(a+c{)}^2+\Delta )} \\ {\sigma }_x=2cd-be \\ {\sigma }_y=2ae-bd \end{gathered}$$

3、二次曲线的几何分类

$$\{\begin{array}{l}\Delta <0,(a+c){\delta }_1\{\begin{array}{l}>0,实椭圆(a=c\ne 0,b=0时为实圆)\\ =0,两条相交的虚直线\\ <0,虚椭圆(a=c\ne 0,b=0时为虚圆)\end{array}\\ \\ \Delta <0,{\delta }_1\{\begin{array}{l}>0,双曲线(a+c=0时为等轴双曲线)\\ =0,两条相交的实直线\\ <0,双曲线(a+c=0时为等轴的双曲线)\end{array}\\ \\ \Delta =0,(a+c){\delta }_1\{\begin{array}{l}>0,抛物线\\ =0,{\sigma }_2\{\begin{array}{l}>0,两条平行的实直线\\ =0,两条重合的实直线\\ <0,两条平行的虚直线\end{array}\\ <0,不存在\end{array}\end{array}$$

4、圆参数计算

• 判别条件：$$\Delta <0,(a+c){\delta }_1>0,a=c\ne 0,b=0时为实圆$$

• 圆心坐标：$$x_0=\frac{{\sigma }_x}{\Delta },y_0=\frac{{\sigma }_y}{\Delta }$$

• 水平方向两端点坐标：

$$\{\begin{array}{l}{x}_{1,2}=x_0\pm b\sqrt{|\frac{{\delta }_1}{\Delta \cdot [{\delta }_2-(a+c)]\cdot {\sigma }_2\cdot [{\sigma }_2+(a-c)]}|}\\ y_{1,2}=y_0\pm \sqrt{|\frac{{\delta }_1\cdot [{\sigma }_2+(a-c)]}{\Delta \cdot [{\sigma }_2-(a+c)]\cdot {\sigma }_2}|}\end{array}$$

5、椭圆参数计算

• 判别条件：$$\Delta <0,(a+c){\delta }_1>0时为椭圆，且a=c\ne 0,b=0时为实圆$$

• 圆心：$$x_0=\frac{{\sigma }_x}{\Delta },y_0=\frac{{\sigma }_y}{\Delta }$$

• 长端两端点坐标:

  • ${\delta }_1>0时:$
    $$长轴\{\begin{array}{l}{x}_{1,2}=x_0\pm b\sqrt{|\frac{{\delta }_1}{\Delta \cdot [{\delta }_2-(a+c)]\cdot {\sigma }_2\cdot [{\sigma }_2+(a-c)]}|}\\ y_{1,2}=y_0\pm \sqrt{|\frac{{\delta }_1\cdot [{\sigma }_2+(a-c)]}{\Delta \cdot [{\sigma }_2-(a+c)]\cdot {\sigma }_2}|}\end{array}$$

  • ${\delta }_1<0时:$
    $$长轴\{\begin{array}{l}{x}_{1,2}=x_0\pm \sqrt{|\frac{{\delta }_1\cdot [{\sigma }_2+(a-c)]}{\Delta \cdot [{\sigma }_2+(a+c)]\cdot {\sigma }_2}|}\\ y_{1,2}=y_0\pm b\sqrt{|\frac{{\delta }_1}{\Delta \cdot [{\delta }_2+(a+c)]\cdot {\sigma }_2\cdot [{\sigma }_2+(a-c)]}|}\end{array}$$

• 短半轴端点坐标：

  • ${\delta }_1>0时:$
    $$短轴\{\begin{array}{l}{x}_{3,4}=x_0\pm \sqrt{|\frac{{\delta }_1\cdot [{\sigma }_2+(a-c)]}{\Delta \cdot [{\sigma }_2+(a+c)]\cdot {\sigma }_2}|}\\ y_{3,4}=y_0\pm b\sqrt{|\frac{{\delta }_1}{\Delta \cdot [{\delta }_2+(a+c)]\cdot {\sigma }_2\cdot [{\sigma }_2+(a-c)]}|}\end{array}$$

  • ${\delta }_1<0时:$
    $$短轴\{\begin{array}{l}{x}_{3,4}=x_0\pm b\sqrt{|\frac{{\delta }_1}{\Delta \cdot [{\delta }_2-(a+c)]\cdot {\sigma }_2\cdot [{\sigma }_2+(a-c)]}|}\\ y_{3,4}=y_0\pm \sqrt{|\frac{{\delta }_1\cdot [{\sigma }_2+(a-c)]}{\Delta \cdot [{\sigma }_2-(a+c)]\cdot {\sigma }_2}|}\end{array}$$

6、双曲线参数计算

• 平面双曲线方程

  • 标准双曲线方程：
    $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$$
    $a$为双曲线实轴的一半，$b$为双曲线虚轴的一半，$c$为焦距的一半。

  • 标准双曲线参数方程：

    $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$$的参数方程为：
    $$\{\begin{array}{l}x=asect\\ y=btant\end{array},t\in [0,2\pi ),t\ne \frac{\pi }{2},t\ne \frac{3\pi }{2}$$

  • 含旋转和平移的双曲线参数方程

    将标准双曲线$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$$的参数方程进行平移$(x_0,y_0)$，旋转$\theta$后
    $$\{\begin{array}{l}x=(asect+x_0)\ast cos\theta +(btant+y_0)\ast sin\theta \\ y=-(asect+x_0)\ast sin\theta +(btant+y_0)\ast cos\theta \end{array}$$

• 二次曲线形式的平面双曲线方程

  • 判别条件：$\Delta <0,{\delta }_1\ne 0$时，为双曲线，且a+c=0时为等轴双曲线

• 长端两端点坐标:

  • ${\delta }_1>0时:$
    $$长轴\{\begin{array}{l}{x}_{1,2}=x_0\pm b\sqrt{|\frac{{\delta }_1}{\Delta \cdot [{\delta }_2-(a+c)]\cdot {\sigma }_2\cdot [{\sigma }_2+(a-c)]}|}\\ y_{1,2}=y_0\pm \sqrt{|\frac{{\delta }_1\cdot [{\sigma }_2+(a-c)]}{\Delta \cdot [{\sigma }_2-(a+c)]\cdot {\sigma }_2}|}\end{array}$$

  • ${\delta }_1<0时:$
    $$长轴\{\begin{array}{l}{x}_{1,2}=x_0\pm \sqrt{|\frac{{\delta }_1\cdot [{\sigma }_2+(a-c)]}{\Delta \cdot [{\sigma }_2+(a+c)]\cdot {\sigma }_2}|}\\ y_{1,2}=y_0\pm b\sqrt{|\frac{{\delta }_1}{\Delta \cdot [{\delta }_2+(a+c)]\cdot {\sigma }_2\cdot [{\sigma }_2+(a-c)]}|}\end{array}$$

• 短半轴端点坐标：

  • ${\delta }_1>0时:$
    $$短轴\{\begin{array}{l}{x}_{3,4}=x_0\pm \sqrt{|\frac{{\delta }_1\cdot [{\sigma }_2+(a-c)]}{\Delta \cdot [{\sigma }_2+(a+c)]\cdot {\sigma }_2}|}\\ y_{3,4}=y_0\pm b\sqrt{|\frac{{\delta }_1}{\Delta \cdot [{\delta }_2+(a+c)]\cdot {\sigma }_2\cdot [{\sigma }_2+(a-c)]}|}\end{array}$$

  • ${\delta }_1<0时:$
    $$短轴\{\begin{array}{l}{x}_{3,4}=x_0\pm b\sqrt{|\frac{{\delta }_1}{\Delta \cdot [{\delta }_2-(a+c)]\cdot {\sigma }_2\cdot [{\sigma }_2+(a-c)]}|}\\ y_{3,4}=y_0\pm \sqrt{|\frac{{\delta }_1\cdot [{\sigma }_2+(a-c)]}{\Delta \cdot [{\sigma }_2-(a+c)]\cdot {\sigma }_2}|}\end{array}$$

7、抛物线参数计算

• 平面抛物线的参数方程：
  $$\begin{gathered} x=(2p{t}^2+x_0)\ast cos\theta +(2pt+y_0)\ast sin\theta \\ y=-(2p{t}^2+x_0)\ast sin\theta +(2pt+y_0)\ast cos\theta \end{gathered}$$
  其中，$p$为焦点掉准线的距离，即焦准距；$\theta$为抛物线的旋转角，即准线与x轴的旋转角。

• 判别条件：$Delta=0,(a+c){\delta }_1>0$

• 顶点坐标：

$$\{\begin{array}{l}{x}_1=\frac{1}{2(a+c)}[\frac{{\sigma }_x}{4}(\frac{{\delta }_2}{{\sigma }_1}+\frac{1}{a+c})-d]\\ y_1=\frac{1}{2(a+c)}[\frac{{\sigma }_y}{4}(\frac{{\delta }_2}{{\sigma }_1}+\frac{1}{a+c})-e]\end{array}$$

• 焦点坐标：

$$\{\begin{array}{l}{x}_5=x_1-\frac{{\sigma }_x}{8(a+c{)}^2}=\frac{1}{2(a+c)}(\frac{{\sigma }_x\cdot {\delta }_2}{4\cdot {\sigma }_1}-d)\\ y_5=y_1-\frac{{\sigma }_y}{8(a+c{)}^2}=\frac{1}{2(a+c)}(\frac{{\sigma }_y\cdot {\delta }_2}{4\cdot {\sigma }_1}-e)\end{array}$$

• 准线方程：
  $${\sigma }_x\cdot [x-(x_1+\frac{{\sigma }_x}{8(a+c{)}^2})]+{\sigma }_x\cdot [y-(y_1+\frac{{\sigma }_y}{8(a+c{)}^2})]=0$$
• 焦准距：
  $$p=\frac{1}{2|a+c|}\cdot \sqrt{\frac{{\sigma }_1}{a+c}}$$

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