## 2016年11月24日木曜日

### 数学 - 2次曲線と直線 - 2次曲線 – 2次曲線と直線(楕円・双曲線と直線(焦点、接線、角))

$\begin{array}{l}|\frac{\frac{{y}_{0}}{{x}_{0}-c}+\frac{{b}^{2}{x}_{0}}{{a}^{2}{y}_{0}}}{1-\frac{{b}^{2}{x}_{0}}{{a}^{2}{y}_{0}}·\frac{{y}_{0}}{{x}_{0}-c}}|\\ =|\frac{\frac{{a}^{2}{y}_{0}{}^{2}+{b}^{2}{x}_{0}\left({x}_{0}-c\right)}{{a}^{2}{y}_{0}\left({x}_{0}-c\right)}}{\frac{{a}^{2}{y}_{0}\left({x}_{0}-c\right)-{b}^{2}{x}_{0}{y}_{0}}{{a}^{2}{y}_{0}\left({x}_{0}-c\right)}}|\\ =|\frac{{a}^{2}{y}_{0}{}^{2}+{b}^{2}{x}_{0}\left({x}_{0}-c\right)}{{a}^{2}{y}_{0}\left({x}_{0}-c\right)-{b}^{2}{x}_{0}{y}_{0}}|\\ =|\frac{{a}^{2}{b}^{2}-{b}^{2}c{x}_{0}}{\left({a}^{2}-{b}^{2}\right){x}_{0}{y}_{0}-{a}^{2}c{y}_{0}}|\\ =|\frac{{a}^{2}{b}^{2}-{b}^{2}c{x}_{0}}{{c}^{2}{x}_{0}{y}_{0}-{a}^{2}c{y}_{0}}|\\ =|\frac{{b}^{2}\left({a}^{2}-c{x}_{0}\right)}{c{y}_{0}\left(c{x}_{0}-{a}^{2}\right)}|\\ =\frac{{b}^{2}}{c{y}_{0}}\\ \\ |\frac{\frac{{y}_{0}}{{x}_{0}+c}+\frac{{b}^{2}{x}_{0}}{{a}^{2}{y}_{0}}}{1-\frac{{y}_{0}}{{x}_{0}+c}·\frac{{b}^{2}{x}_{0}}{{a}^{2}{y}_{0}}}|\\ =|\frac{\frac{{a}^{2}{y}_{0}{}^{2}+{b}^{2}{x}_{0}\left({x}_{0}+c\right)}{{a}^{2}{y}_{0}\left({x}_{0}+c\right)}}{\frac{{a}^{2}{y}_{0}\left({x}_{0}+c\right)-{b}^{2}{x}_{0}{y}_{0}}{{a}^{2}{y}_{0}\left({x}_{0}+c\right)}}|\\ =|\frac{{a}^{2}{y}_{0}{}^{2}+{b}^{2}{x}_{0}\left({x}_{0}+c\right)}{{a}^{2}{y}_{0}\left({x}_{0}+c\right)-{b}^{2}{x}_{0}{y}_{0}}|\\ =|\frac{{a}^{2}{b}^{2}+{b}^{2}c{x}_{0}}{{c}^{2}{x}_{0}{y}_{0}+{a}^{2}c{y}_{0}}|\\ =|\frac{{b}^{2}\left({a}^{2}+c{x}_{0}\right)}{c{y}_{0}\left(c{x}_{0}+{a}^{2}\right)}|\\ =\frac{{b}^{2}}{c{y}_{0}}\end{array}$