HSC 扩展 1 和 2 数学/4 个单元/圆锥曲线

椭圆

椭圆切线：笛卡尔方法

椭圆的笛卡尔方程为 ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$。求梯度时微分（使用隐式微分来简化过程）

双曲线

双曲线切线：笛卡尔方法

双曲线的笛卡尔方程为 ${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1$。求梯度时微分（使用隐式微分来简化过程）

${\begin{aligned}0&={\frac {2x}{a^{2}}}-{\frac {2y}{b^{2}}}{\frac {dy}{dx}}\\{\frac {dy}{dx}}{\frac {y}{b^{2}}}&={\frac {x}{a^{2}}}\\{\frac {dy}{dx}}&={\frac {b^{2}}{a^{2}}}\times {\frac {x}{y}}\end{aligned}}$

然后我们可以将其代入我们的点斜式， $y-y_{1}=m(x-x_{1})$，使用点 $P(x_{1},y_{1})$

在 $P$， $m={\frac {b^{2}}{a^{2}}}{\frac {x_{1}}{y_{1}}}$。

${\begin{aligned}y-y_{1}&={\frac {b^{2}}{a^{2}}}{\frac {x_{1}}{y_{1}}}(x-x_{1})\\yy_{1}-y_{1}^{2}&={\frac {b^{2}}{a^{2}}}x_{1}(x-x_{1})\\{\frac {yy_{1}}{b^{2}}}-{\frac {y_{1}^{2}}{b^{2}}}&={\frac {x_{1}}{a^{2}}}(x-x_{1})\\{\frac {yy_{1}}{b^{2}}}-{\frac {y_{1}^{2}}{b^{2}}}&={\frac {xx_{1}}{a^{2}}}-{\frac {x_{1}^{2}}{a^{2}}}\\{\frac {x_{1}^{2}}{a^{2}}}-{\frac {y_{1}^{2}}{b^{2}}}&={\frac {xx_{1}}{a^{2}}}-{\frac {yy_{1}}{b^{2}}}\\\end{aligned}}$

但我们知道，根据双曲线的定义，${\frac {x_{1}^{2}}{a^{2}}}-{\frac {y_{1}^{2}}{b^{2}}}=1$，所以

${\frac {xx_{1}}{a^{2}}}-{\frac {yy_{1}}{b^{2}}}=1$

双曲线法线：笛卡尔方法

法线的斜率由$-{\frac {dx}{dy}}$给出，即$-{\frac {a^{2}}{b^{2}}}\times {\frac {y}{x}}$。求出方程，

${\begin{aligned}y-y_{1}&=-{\frac {a^{2}}{b^{2}}}{\frac {y_{1}}{x_{1}}}(x-x_{1})\\yb^{2}-y_{1}b^{2}&=-a^{2}{\frac {y_{1}}{x_{1}}}(x-x_{1})\\{\frac {b^{2}y}{y_{1}}}-b^{2}&=-{\frac {a^{2}}{x_{1}}}(x-x_{1})\\{\frac {b^{2}y}{y_{1}}}-b^{2}&=-{\frac {a^{2}x}{x_{1}}}+a^{2}\\{\frac {a^{2}x}{x_{1}}}+{\frac {b^{2}y}{y_{1}}}&=a^{2}+b^{2}\end{aligned}}$