每日一道選填題4（2022乙卷圓錐曲線）

2022-09-06 21:50 作者:因戀相思久 0人讀過 | 我要投稿

5.設 $F$ 為拋物線 $C%EF%BC%9Ay%5E2%20%3D4x$ 的 $%E7%84%A6%E7%82%B9%EF%BC%8C%E7%82%B9A%E5%9C%A8C%E4%B8%8A%EF%BC%8C%E7%82%B9B%EF%BC%883%2C0%EF%BC%89%EF%BC%8C$ $%E8%8B%A5%5Cvert%20AF%20%5Cvert%3D%20%5Cvert%20BF%20%5Cvert%20%EF%BC%8C%E5%88%99%5Cvert%20AB%20%5Cvert%20%3D%EF%BC%88%EF%BC%89$

$a.2$

$b.2%5Csqrt%7B2%7D%20$

$c.3$

$d.3%5Csqrt%7B2%7D%20$

$%E6%98%BE%E7%84%B6p%3D2%EF%BC%8CF%EF%BC%881%2C0%EF%BC%89$

$%E6%89%80%E4%BB%A5%5Cvert%20AF%20%5Cvert%3D%20%5Cvert%20BF%20%5Cvert%20%3D2$

$%E5%9B%A0%E4%B8%BAA%E5%9C%A8C%E4%B8%8A%EF%BC%8C%E6%A0%B9%E6%8D%AE%E6%8A%9B%E7%89%A9%E7%BA%BF%E5%AE%9A%E4%B9%89%E6%9C%89$

$x_%7BA%7D%20%2B%5Cfrac%7Bp%7D%7B2%7D%20%3D2$

$x_%7BA%7D%20%3D1%2Cy_%7BA%7D%20%3D%5Cpm%202$

$%5Cvert%20AB%20%5Cvert%20%3D%5Csqrt%7B(x_%7BA%7D-3)%5E2%2By_%7BA%7D%5E2%20%20%7D%20%3D2%5Csqrt%7B2%7D%20$

$%E9%80%89b$

技巧點：

拋物線定義

11.雙曲線 $C$ 的兩個焦點 $F_%7B1%7D%EF%BC%8CF_%7B2%7D%20$ ，以 $C%0A$ 的實軸為直徑的圓記為 $D$ ，過 $F_%7B1%7D%20$ 作 $D$ 的切線與 $C$ 交于 $M%EF%BC%8CN%E4%B8%A4%E7%82%B9%EF%BC%8C%E4%B8%94cos%E2%88%A0F_%7B1%7DN%20F_%7B2%7D%20%3D%5Cfrac%7B3%7D%7B5%7D%20$ ，則 $C$ 的離心率為（）

$a.%5Cfrac%7B%5Csqrt%7B5%7D%20%7D%7B2%7D%20$

$b.%5Cfrac%7B3%7D%7B2%7D%20$

$c.%5Cfrac%7B%5Csqrt%7B13%7D%20%7D%7B2%7D%20$

$d.%5Cfrac%7B%5Csqrt%7B17%7D%20%7D%7B2%7D%20$

$%E8%AE%BE%E5%9C%86%E4%B8%8E%E7%9B%B4%E7%BA%BFMN%E5%88%87%E7%82%B9%E4%B8%BAQ%EF%BC%8C%E4%BD%9CF_%7B2%7D%20H%E2%8A%A5MN%EF%BC%8C%E5%9E%82%E8%B6%B3%E4%B8%BAH$

① $M%EF%BC%8CN%E5%90%8C%E6%94%AF%E6%97%B6$

$%5Cvert%20OQ%20%5Cvert%3Da%EF%BC%8C%20%5Cvert%20OF_%7B1%7D%20%20%5Cvert%20%3Dc$

$%E6%89%80%E4%BB%A5%5Cvert%20QF_%7B1%7D%20%20%5Cvert%20%3Db$

$%E5%9B%A0%E4%B8%BA%E2%96%B3OQF_%7B1%7D%20%E2%88%BD%E2%96%B3F_%7B2%7D%20HF_%7B1%7D%20$

$%E6%89%80%E4%BB%A5%5Cvert%20HF_%7B1%7D%20%20%5Cvert%20%3D2b%EF%BC%8C%5Cvert%20HF_%7B2%7D%20%20%5Cvert%20%3D2a$

$cos%E2%88%A0F_%7B1%7DN%20F_%7B2%7D%20%3D%5Cfrac%7B3%7D%7B5%7D%20%EF%BC%8Csin%E2%88%A0F_%7B1%7DN%20F_%7B2%7D%20%3D%5Cfrac%7B4%7D%7B5%7D%20$

$%E8%A7%A3%E5%BE%97%5Cvert%20NH%20%5Cvert%20%3D%5Cfrac%7B3a%7D%7B2%7D%EF%BC%8C%20%5Cvert%20F_%7B2%7D%20N%20%5Cvert%20%3D%5Cfrac%7B5a%7D%7B2%7D%20$

$%E6%A0%B9%E6%8D%AE%E5%8F%8C%E6%9B%B2%E7%BA%BF%E5%AE%9A%E4%B9%89$

$%20%5Cvert%20F_%7B2%7D%20N%20%5Cvert%20-%20%5Cvert%20F_%7B1%7D%20N%20%5Cvert%20%3D%20%5Cvert%20F_%7B2%7D%20N%20%5Cvert%20-%EF%BC%88%20%5Cvert%20%20NH%20%5Cvert%20-%20%5Cvert%20F_%7B1%7D%20H%20%5Cvert%20%EF%BC%89%3D2a$

$%E5%8D%B3a%2B2b%3D2a$

$%5Cfrac%7Bb%7D%7Ba%7D%3D%20%5Cfrac%7B1%7D%7B2%7D%20$

$e%3D%5Csqrt%7B1%2B%EF%BC%88%5Cfrac%7Bb%7D%7Ba%7D%EF%BC%89%5E2%20%20%7D%20%3D%5Cfrac%7B%5Csqrt%7B5%7D%20%7D%7B2%7D%20$

② $M%2CN%E5%BC%82%E6%94%AF%E6%97%B6$