每日一道選填題2（圓錐曲線）

2022-09-04 16:46 作者:因戀相思久 0人讀過 | 我要投稿

$7.%E5%B7%B2%E7%9F%A5%E7%82%B9M%E5%9C%A8%E6%8A%9B%E7%89%A9%E7%BA%BFC%EF%BC%9Ay%5E2%3D2px%EF%BC%88p%EF%BC%9E0%EF%BC%89$ 上，若以點 $M$ 為圓心半徑為5的圓與拋物線 $C$ 的準線相切，且與 $x$ 軸相交的弦長為6，則 $p$ 為（）

$a.2$

$b.8$

$c.2%E6%88%968$

$d.6$

$%E5%9B%A0%E4%B8%BAM%E5%9C%A8%E6%8A%9B%E7%89%A9%E7%BA%BF%E4%B8%8A$

$%E4%BB%A5M%E4%B8%BA%E5%9C%86%E4%B8%8E%E5%87%86%E7%BA%BF%E7%9B%B8%E5%88%87%EF%BC%8C%E4%B8%94%E4%B8%8Ex%E8%BD%B4%E7%9B%B8%E4%BA%A4$

$%E8%AE%BE%E4%BA%A4%E7%82%B9%E4%B8%BAA%EF%BC%8CB$

$%E6%A0%B9%E6%8D%AE%E6%8A%9B%E7%89%A9%E7%BA%BF%E5%AE%9A%E4%B9%89$

①以 $A$ 為焦點

$p%2B3%3D5$

$p%3D2$

②以 $B$ 為焦點

$p-3%3D5$

$p%3D8$

$%E6%95%85%E9%80%89c$

技巧點：

拋物線定義

$16.%E5%B7%B2%E7%9F%A5%E5%8F%8C%E6%9B%B2%E7%BA%BFC%EF%BC%9A%5Cfrac%7Bx%5E2%20%7D%7B6%7D-%20%5Cfrac%7By%5E2%20%7D%7B3%7D%20%3D1$ 的右焦點為 $F$ ，過點 $F$ 斜率為 $k$ 的 $%E7%9B%B4%E7%BA%BFl%E4%B8%8E%E5%8F%8C%E6%9B%B2%E7%BA%BFC%E7%9A%84%E5%8F%B3%E6%94%AF%E4%BA%A4%E4%BA%8EA%EF%BC%8CB$ 兩點，點 $P%EF%BC%88-2%5Csqrt%7B3%7D%20%2C0%EF%BC%89$ ，若 $%E2%96%B3ABP$ 的外心 $Q$ 的橫坐標為0，則直線 $l$ 的方程為 $%5Cotimes%20$

$%E6%98%93%E7%9F%A5%E2%96%B3ABP%E5%A4%96%E6%8E%A5%E5%9C%86%E7%9A%84%E5%9C%86%E5%BF%83%E4%B8%BAM%E4%B8%94%E5%9C%A8y%E8%BD%B4%E4%B8%8A$

$%E8%AE%BEP%E5%85%B3%E4%BA%8Ey%E8%BD%B4%E7%9A%84%E5%AF%B9%E7%A7%B0%E7%82%B9%E4%B8%BAP_%7B1%7D%20$

$%E6%A0%B9%E6%8D%AE%E5%9C%86%E7%9A%84%E5%AF%B9%E7%A7%B0%E6%80%A7$

$P_%7B1%7D%20%E4%B9%9F%E5%9C%A8%E5%9C%86%E4%B8%8A$

$%E5%9B%A0%E6%AD%A4A%EF%BC%8CP%EF%BC%8CB%EF%BC%8CP_%7B1%7D%20%E5%85%B1%E5%9C%86$

$%E4%B8%8D%E5%A6%A8%E5%BD%93%5Cvert%20AF%20%5Cvert%3D%5Cfrac%7Bb%5E2%7D%7Ba-ccos%5Ctheta%20%7D%20%EF%BC%88%5Ctheta%E4%B8%BA%E5%80%BE%E6%96%9C%E8%A7%92%20%EF%BC%89$

$%E5%88%99$ $%5Cvert%20BF%20%5Cvert%3D%5Cfrac%7Bb%5E2%7D%7Ba%2Bccos%5Ctheta%20%7D%20$

$%5Cvert%20PF%20%5Cvert%20%3D2%5Csqrt%7B3%7D%20%2B3$

$%5Cvert%20P_%7B1%7D%20F%20%5Cvert%20%3D2%5Csqrt%7B3%7D%20-3$

$%E6%A0%B9%E6%8D%AE%E7%9B%B8%E4%BA%A4%E5%BC%A6%E5%AE%9A%E7%90%86%E6%9C%89$

$%5Cvert%20AF%20%5Cvert%20%5Cvert%20BF%20%5Cvert%20%3D%5Cvert%20PF%20%5Cvert%20%5Cvert%20P_%7B1%7D%20F%20%5Cvert%20$

$%5Cfrac%7Bb%5E4%7D%7Ba%5E2-c%5E2cos%5E2%5Ctheta%20%7D%20%3D%EF%BC%882%5Csqrt%7B3%7D%20%2B3%EF%BC%89%EF%BC%882%5Csqrt%7B3%7D%20-3%EF%BC%89$

$%5Cfrac%7B9%7D%7B6-9cos%5E2%5Ctheta%20%20%7D%20%3D3$

$%E5%8D%B3cos%5E2%5Ctheta%20%20%3D%5Cfrac%7B1%7D%7B3%7D%20%EF%BC%8Csin%5E2%5Ctheta%20%20%3D%5Cfrac%7B2%7D%7B3%7D%20$

$tan%5E2%5Ctheta%20%20%3D2$

$k_%7B1%7D%20%3D%5Csqrt%7B2%7D%EF%BC%8C%20k_%7B2%7D%20%3D-%5Csqrt%7B2%7D%20$

$%E7%9B%B4%E7%BA%BFl%E6%96%B9%E7%A8%8B%E4%B8%BA%5Csqrt%7B2%7D%20x-y-3%5Csqrt%7B2%7D%20%3D0%E6%88%96%5Csqrt%7B2%7D%20x%2By-3%5Csqrt%7B2%7D%20%3D0$

技巧點：

①

$%5Cvert%20AF%20%5Cvert%3D%5Cfrac%7Bb%5E2%7D%7Ba-ccos%5Ctheta%20%7D%EF%BC%88%E9%95%BF%E7%9A%84%EF%BC%89%EF%BC%8C%20%5Cvert%20BF%20%5Cvert%3D%5Cfrac%7Bb%5E2%7D%7Ba%2Bccos%5Ctheta%20%7D%20%EF%BC%88%E7%9F%AD%E7%9A%84%EF%BC%89$