每日一道選填題6（圓錐曲線）

2022-09-12 17:38 作者:因戀相思久 0人讀過 | 我要投稿

$11.%E5%B7%B2%E7%9F%A5%E6%8A%9B%E7%89%A9%E7%BA%BFC%EF%BC%9Ax%5E2%3D12y%E7%9A%84%E7%84%A6%E7%82%B9%E4%B8%BAF$ ，其準(zhǔn)線與y軸的交點(diǎn)為A，點(diǎn)B為拋物線 $C$ 上一動(dòng)點(diǎn)，當(dāng) $%5Cfrac%7B%5Cvert%20AB%20%5Cvert%20%7D%7B%5Cvert%20FB%20%5Cvert%20%7D%20$ 取得最大值時(shí)，直線AB的傾斜角為（）

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

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

$c.%5Cfrac%7B%5Cpi%20%7D%7B6%7D%20%E6%88%96%5Cfrac%7B5%5Cpi%20%7D%7B6%7D%20$

$d.%5Cfrac%7B%5Cpi%20%7D%7B4%7D%E6%88%96%20%5Cfrac%7B3%5Cpi%20%7D%7B4%7D%20$

$%E6%98%BE%E7%84%B6p%3D6%EF%BC%8CF%EF%BC%880%2C3%EF%BC%89%EF%BC%8CA%EF%BC%880%EF%BC%8C-3%EF%BC%89$

$%E8%AE%BEB%EF%BC%88x_%7B0%7D%20%EF%BC%8Cy_%7B0%7D%20%EF%BC%89$

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

$%5Cvert%20FB%20%5Cvert%20%3Dy_%7B0%7D%20%2B3$

$%5Cfrac%7B%5Cvert%20AB%20%5Cvert%20%7D%7B%5Cvert%20FB%20%5Cvert%20%7D%20%3D%5Cfrac%7B%5Csqrt%7Bx_%7B0%7D%20%5E2%2B%EF%BC%88y_%7B0%7D%20%2B3%EF%BC%89%5E2%7D%20%7D%7By_%7B0%7D%20%2B3%7D%20$

$%5Csqrt%7B%5Cfrac%7Bx_%7B0%7D%5E2%2B%EF%BC%88y_%7B0%7D%20%2B3%EF%BC%89%5E2%20%20%20%7D%7B%EF%BC%88y_%7B0%7D%20%2B3%EF%BC%89%5E2%7D%20%7D%20$

$%5Csqrt%7B%5Cfrac%7B12y_%7B0%7D%20%7D%7By_%7B0%7D%20%5E2%2B6y_%7B0%7D%20%2B9%7D%2B1%20%7D%20$

$%5Csqrt%7B%5Cfrac%7B12%20%7D%7By_%7B0%7D%20%2B6%20%2B%5Cfrac%7B9%7D%7By_%7B0%7D%20%7D%20%7D%2B1%20%7D%20%5Cleq%20%5Csqrt%7B2%7D%20$

$%E5%BD%93%E4%B8%94%E4%BB%85%E5%BD%93x_%7B0%7D%20%3D%5Cpm6%20%EF%BC%8C%2Cy_%7B0%7D%20%3D3%E6%97%B6%E6%88%90%E7%AB%8B$

$k_%7BAB%7D%20%3D%5Cfrac%7B-3-3%20%7D%7B0%5Cpm6%20%7D%20%3D%5Cpm%201$

$%5Ctheta%20%3D%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%E6%88%96%5Cfrac%7B3%5Cpi%20%7D%7B4%7D%20%EF%BC%8C%E9%80%89d$

技巧點(diǎn)：

拋物線定義，基本不等式

$16.%E8%AE%BE%E5%8F%8C%E6%9B%B2%E7%BA%BF%5CGamma%20%EF%BC%9A%5Cfrac%7Bx%5E2%20%7D%7Ba%5E2%20%7D%20-%5Cfrac%7By%5E2%20%7D%7Bb%5E2%20%7D%20%3D1%EF%BC%88a%EF%BC%9E0%EF%BC%8Cb%EF%BC%9E0%EF%BC%89$ $%E7%9A%84%E5%B7%A6%EF%BC%8C%E5%8F%B3%E7%84%A6%E7%82%B9%E4%B8%BAF_%7B1%7D%20%EF%BC%88-c%EF%BC%8C0%EF%BC%89$ $%EF%BC%8CF_%7B2%7D%20%EF%BC%88c%EF%BC%8C0%EF%BC%89%EF%BC%8C%E8%BF%87%E7%82%B9P%EF%BC%88-2c%EF%BC%8C0%EF%BC%89$ 且斜率為 $%5Cfrac%7B1%7D%7B2%7D%20$ 的直線與雙曲線的左，右兩支分別交于M，N兩點(diǎn)， $%E8%8B%A5%5Cvert%20PN%20%5Cvert%20%3D3%5Cvert%20PM%20%5Cvert%20$ ，且直線 $F_%7B2%7D%20N$ 的斜率為3，則 $%5CGamma%20$ 的離心率為 $%5Cast%20%5Cast%20$

$%E6%A0%B9%E6%8D%AE%E9%A2%98%E6%84%8F%E7%9F%A5%5Cvert%20PF_%7B2%7D%20%20%5Cvert%3D3%20%5Cvert%20PF_%7B1%7D%20%20%5Cvert%20$

$%E6%89%80%E4%BB%A5%E2%96%B3F_%7B1%7D%20MP%E2%88%BD%E2%96%B3F_%7B2%7D%20NP$

$%E5%8D%B3F_%7B1%7D%20M%E2%88%A5F_%7B2%7D%20N%2C%5Cvert%20MF_%7B1%7D%5Cvert%20%20%3D3%5Cvert%20NF_%7B2%7D%20%5Cvert%20$

$%E5%BB%B6%E9%95%BFMF_%7B1%7D%20%E4%BA%A4%E5%8F%8C%E6%9B%B2%E7%BA%BF%E4%BA%8EQ$

$%E6%A0%B9%E6%8D%AE%E5%8F%8C%E6%9B%B2%E7%BA%BF%E7%9A%84%E5%AF%B9%E7%A7%B0%E6%80%A7%E5%8F%AF%E7%9F%A5$

$%5Cvert%20QF_%7B1%7D%5Cvert%20%20%3D%5Cvert%20NF_%7B2%7D%20%5Cvert%20$

$%E6%89%80%E4%BB%A53%5Cvert%20QF_%7B1%7D%20%20%5Cvert%20%3D%5Cvert%20MF_%7B1%7D%20%20%5Cvert%20$

$k_%7BNF_%7B2%7D%20%7D%20%3Dk_%7BMQ%20%7D%3D3$

$tan%5Ctheta%20%3D3%EF%BC%88%5Ctheta%20%E4%B8%BAMQ%E5%80%BE%E6%96%9C%E8%A7%92%EF%BC%89$

$%E6%89%80%E4%BB%A5cos%5Ctheta%20%3D%5Cfrac%7B%5Csqrt%7B10%7D%20%7D%7B10%7D%20$

$e%5Cvert%20cos%5Ctheta%20%20%5Cvert%20%3D%5Cvert%20%5Cfrac%7B%5Clambda%20-1%7D%7B%5Clambda%2B1%20%7D%20%20%5Cvert%20%EF%BC%88%5Clambda%20%E4%B8%BA%E7%84%A6%E5%8D%8A%E5%BE%84%E4%B9%8B%E6%AF%94%EF%BC%89$

$%E6%98%BE%E7%84%B6%5Clambda%20%3D3$

$%E6%89%80%E4%BB%A5e%3D%5Cfrac%7B%5Csqrt%7B10%7D%20%7D%7B2%7D%20$

技巧點(diǎn)：

雙曲線的對(duì)稱性，相似三角形，焦半徑公式

附：

$%E5%BC%82%E6%94%AFe%5Cvert%20cos%5Ctheta%20%20%5Cvert%20%3D%5Cvert%20%5Cfrac%7B%5Clambda%20%2B1%7D%7B%5Clambda-1%20%7D%20%20%5Cvert%20%EF%BC%88%5Clambda%20%E4%B8%BA%E7%84%A6%E5%8D%8A%E5%BE%84%E4%B9%8B%E6%AF%94%EF%BC%89$

$%E5%90%8C%E6%94%AFe%5Cvert%20cos%5Ctheta%20%20%5Cvert%20%3D%5Cvert%20%5Cfrac%7B%5Clambda%20-1%7D%7B%5Clambda%2B1%20%7D%20%20%5Cvert%20%EF%BC%88%5Clambda%20%E4%B8%BA%E7%84%A6%E5%8D%8A%E5%BE%84%E4%B9%8B%E6%AF%94%EF%BC%89$