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每日一道選填題（圓錐曲線）

2022-09-03 21:29 作者:因戀相思久 0人讀過 | 我要投稿

$7.%E5%B7%B2%E7%9F%A5F_%7B1%7D%20%EF%BC%8CF_%7B2%7D%20%E5%88%86%E5%88%AB%E4%B8%BA%E6%A4%AD%E5%9C%86%5Cfrac%7Bx%5E2%20%7D%7B4%7D%2B%20%5Cfrac%7By%5E2%20%7D%7B2%7D%20%3D1$ 的左右焦點(diǎn)，點(diǎn) $P$ 為橢圓上一點(diǎn)，過 $F_%7B2%7D%20$ 為圓心的圓與直線 $PF_%7B1%7D%20$ 恰好相切于點(diǎn) $P$ ，則 $%E2%88%A0PF_%7B1%7D%20F_%7B2%7D%20$ 是（）

a. $%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20$

b. $%5Cfrac%7B%5Cpi%20%7D%7B6%7D%20$

c. $%5Cfrac%7B5%5Cpi%20%7D%7B12%7D%20$

d. $%5Cfrac%7B%5Cpi%20%7D%7B3%7D%20$

$%E5%9B%A0%E4%B8%BA$

$b%3Dc%3D%5Csqrt%7B2%7D%20%EF%BC%8Ca%3D2$

$%E5%9C%86%E4%B8%8E%E7%9B%B4%E7%BA%BF%E7%9B%B8%E5%88%87%EF%BC%8C%E5%BE%97%E5%88%B0%E2%88%A0F_%7B1%7D%20PF_%7B2%7D%20%3D%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20$

$P%E5%88%B0x%E8%BD%B4%E8%B7%9D%E7%A6%BB%E4%B8%BAh$

$S_%7B%E2%96%B3F_%7B1%7D%20PF_%7B2%7D%20%7D%20%3Db%5E2tan%5Cfrac%7B%E2%88%A0F_%7B1%7D%20PF_%7B2%7D%20%7D%7B2%7D%20%3D%5Cfrac%7B1%7D%7B2%7D%202ch$

$%E8%A7%A3%E5%BE%97h%3Db%3D%5Csqrt%7B2%7D%20$

$%E5%9B%A0%E6%AD%A4P%E5%9C%A8%E4%B8%8A%E6%88%96%E4%B8%8B%E9%A1%B6%E7%82%B9$

$tan%E2%88%A0PF_%7B1%7D%20F_%7B2%7D%20%3D%5Cfrac%7Bb%7D%7Bc%7D%20%3D1$

$%E2%88%A0PF_%7B1%7D%20F_%7B2%7D%20%3D%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%EF%BC%8C%E9%80%89%E6%8B%A9a$

技巧點(diǎn)：

$%E7%84%A6%E7%82%B9%E4%B8%89%E8%A7%92%E5%BD%A2S_%7B%E2%96%B3F_%7B1%7D%20PF_%7B2%7D%20%7D%20%3Db%5E2tan%5Cfrac%7B%E2%88%A0F_%7B1%7D%20PF_%7B2%7D%20%7D%7B2%7D$

15.已知拋物線 $y%5E2%3D8x$ 的焦點(diǎn) $F%0A$ ，過點(diǎn) $F%0A$ 的直線交拋物線于 $A%EF%BC%8CB%EF%BC%8C$ 兩點(diǎn)，延長(zhǎng) $FB$ 交準(zhǔn)線于點(diǎn) $C$ ，分別過點(diǎn) $A%EF%BC%8CB$ 作準(zhǔn)線的垂線，垂足分別記為 $M%EF%BC%8CN$ ，若 $%5Cvert%20BC%20%5Cvert%3D2%20%5Cvert%20BN%20%5Cvert%20$ ，則 $%E2%96%B3AFM$ 的面積為 $%5Cotimes%20$

$p%3D4$

$cos%E2%88%A0CBM%3Dcos%E2%88%A0FAM%3D%5Cfrac%7B%5Cvert%20NB%20%5Cvert%20%7D%7B%5Cvert%20BC%20%5Cvert%20%7D%20%3D%5Cfrac%7B1%7D%7B2%7D%20$

$%E2%88%A0FAM%3D%5Cfrac%7B%5Cpi%20%7D%7B3%7D%20%EF%BC%8C%E7%9B%B4%E7%BA%BFAB%E5%80%BE%E6%96%9C%E8%A7%92%5Ctheta%20%3D%5Cfrac%7B%5Cpi%20%7D%7B3%7D$

$%E6%89%80%E4%BB%A5%5Cvert%20AF%20%5Cvert%20%3D%5Cfrac%7Bp%7D%7B1-cos%5Ctheta%20%7D%20%3D8$

由拋物線定義有

$%5Cvert%20AM%20%5Cvert%3D%20%5Cvert%20AF%5Cvert%20%3D8$

$S_%7B%E2%96%B3AFM%7D%3D%5Cfrac%7B1%7D%7B2%7D%20%5Cvert%20AM%20%5Cvert%20%5Cvert%20AF%20%5Cvert%20sin%5Cfrac%7B%5Cpi%20%7D%7B3%7D%20%3D16%5Csqrt%7B3%7D%20$

技巧點(diǎn)：

$%5Cvert%20AF%20%5Cvert%20%3D%5Cfrac%7Bp%7D%7B1-cos%5Ctheta%20%7D%20%EF%BC%88%E9%95%BF%E7%9A%84%EF%BC%89%EF%BC%8C%5Cvert%20BF%20%5Cvert%20%3D%5Cfrac%7Bp%7D%7B1%2Bcos%5Ctheta%20%7D%20%EF%BC%88%E7%9F%AD%E7%9A%84%EF%BC%89$