每日一道选填题7（2022新高考1卷圆锥曲线）

2022-09-29 13:28 作者:因恋相思久 0人读过 | 我要投稿

$%E5%B7%B2%E7%9F%A5O%E4%B8%BA%E5%9D%90%E6%A0%87%E5%8E%9F%E7%82%B9%EF%BC%8C%E7%82%B9A%EF%BC%881%2C1%EF%BC%89$ $%E5%9C%A8%E6%8A%9B%E7%89%A9%E7%BA%BFC%EF%BC%9Ax%5E2%3D2py%EF%BC%88p%EF%BC%9E0%EF%BC%89$ 上，过 $B%EF%BC%880%EF%BC%8C-1%EF%BC%89$ 的直线交 $C%E4%BA%8EP%EF%BC%8CQ$ 两点，则（）

$a.%20C%E7%9A%84%E5%87%86%E7%BA%BF%E4%B8%BAy%3D-1$

$b.%E7%9B%B4%E7%BA%BFAB%E4%B8%8EC%E7%9B%B8%E5%88%87$

$c.%5Cvert%20OP%20%5Cvert%20%5Cvert%20OQ%20%5Cvert%20%EF%BC%9E%5Cvert%20OA%20%5Cvert%20%5E2$

$d.%5Cvert%20BP%20%5Cvert%20%5Cvert%20BQ%20%5Cvert%20%EF%BC%9E%5Cvert%20BA%20%5Cvert%20%5E2$

$%E7%82%B9A%EF%BC%881%EF%BC%8C1%EF%BC%89%E5%9C%A8x%5E2%3D2py%E4%B8%8A%20$

$%E8%A7%A3%E5%BE%97p%3D%5Cfrac%7B1%7D%7B2%7D%20$

$%E6%89%80%E4%BB%A5C%E7%9A%84%E5%87%86%E7%BA%BF%E4%B8%BAy%3D-%5Cfrac%7Bp%7D%7B2%7D%20%3D-%5Cfrac%7B1%7D%7B4%7D%20%EF%BC%8Ca%E9%94%99$

对于b

$k_%7BAB%7D%20%3D%5Cfrac%7B1-%EF%BC%88-1%EF%BC%89%7D%7B1-0%7D%20%3D2$

$f'(x)%3Dy'%3D2x$

$f'(1)%3Dk_%7BAB%7D%20%3D2$

b正确

对于c

$%E8%AE%BEP(x_%7B1%7D%20%2Cy_%7B1%7D%20)%2CQ(x_%7B2%7D%2C%20y_%7B2%7D%20)$

$%E6%89%80%E4%BB%A5%E7%9B%B4%E7%BA%BFPQ%E4%B8%BA%EF%BC%9Ay%3D%EF%BC%88x_%7B1%7D%2B%20x_%7B2%7D%EF%BC%89x-%20x_%7B1%7D%20x_%7B2%7D%20$

$%E7%9B%B4%E7%BA%BFPQ%E8%BF%87B%EF%BC%880%EF%BC%8C-1%EF%BC%89$

$%E6%89%80%E4%BB%A5x_%7B1%7D%20x_%7B2%7D%3D1%EF%BC%8C%20y_%7B1%7D%20y_%7B2%7D%20%3D1$

$%5Cvert%20OP%20%5Cvert%20%5Cvert%20OQ%20%5Cvert%3D%5Csqrt%7B(x_%7B1%7D%20%5E2%2By_%7B1%7D%5E2)(%20x_%7B2%7D%5E2%2B%20y_%7B2%7D%5E2)%20%7D%20$

$%5Csqrt%7B(x_%7B1%7D%20%5E2%2By_%7B1%7D%5E2)(%20x_%7B2%7D%5E2%2B%20y_%7B2%7D%5E2)%20%7D%20%EF%BC%9Ex_%7B1%7D%20x_%7B2%7D%2B%20y_%7B1%7D%20y_%7B2%7D%3D2%3D%20%5Cvert%20OA%20%5Cvert%20%5E2%0A$

$%EF%BC%88%E6%9F%AF%E8%A5%BF%E4%B8%8D%E7%AD%89%E5%BC%8F%EF%BC%89%E4%BA%A6%E5%8F%AF%E4%BB%A5%EF%BC%88%E5%9F%BA%E6%9C%AC%E4%B8%8D%E7%AD%89%E5%BC%8F%EF%BC%89%E5%8C%96%E7%AE%80%E5%81%9A$

$%E6%89%80%E4%BB%A5%5Cvert%20OP%20%5Cvert%20%5Cvert%20OQ%20%5Cvert%20%EF%BC%9E%5Cvert%20OA%20%5Cvert%20%5E2$

$c%E6%AD%A3%E7%A1%AE$

对于d

$%E8%AE%BEAB%E5%80%BE%E6%96%9C%E8%A7%92%E4%B8%BA%5Ctheta%20%EF%BC%8CPQ%E5%80%BE%E6%96%9C%E8%A7%92%E4%B8%BA%5Calpha%20$

$%E4%B8%8D%E5%A6%A8%E4%BB%A40%EF%BC%9C%5Ctheta%20%EF%BC%9C%5Calpha%EF%BC%9C%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%20$

$%5Cfrac%7B1%7D%7Bcos%5E2%5Ctheta%20%7D%20%EF%BC%9C%5Cfrac%7B1%7D%7Bcos%5E2%5Calpha%20%7D%20$

$%5Cvert%20BP%20%5Cvert%20%5Cvert%20BQ%20%5Cvert%20%3D%5Cfrac%7Bx_%7B1%7Dx_%7B2%7D%20%20%7D%7Bcos%5E2%5Calpha%20%7D%3D%20%5Cfrac%7B1%20%20%7D%7Bcos%5E2%5Calpha%20%7D%20$

$%5Cvert%20AB%5Cvert%5E2%20%3D%5Cfrac%7B1%7D%7Bcos%5E2%5Ctheta%20%7D%20$

$%5Cvert%20BP%20%5Cvert%20%5Cvert%20BQ%20%5Cvert%20%EF%BC%9E%5Cvert%20BA%20%5Cvert%20%5E2$

$d%E6%AD%A3%E7%A1%AE$

$%E9%80%89%E6%8B%A9bcd$

技巧点：

$%E6%8A%9B%E7%89%A9%E7%BA%BF%E7%9A%84%E5%AE%9A%E4%B9%89%EF%BC%8C%E4%B8%8D%E7%AD%89%E5%BC%8F%E7%9A%84%E8%BF%90%E7%AE%97%EF%BC%8C%E4%B8%89%E7%82%B9%E5%85%B1%E7%BA%BF%E5%85%B3%E7%B3%BB%EF%BC%8C%E4%B8%8E%E5%AF%BC%E6%95%B0%E7%9A%84%E8%81%94%E7%B3%BB$

$16.%E5%B7%B2%E7%9F%A5%E6%A4%AD%E5%9C%86C%EF%BC%9A%5Cfrac%7Bx%5E2%20%7D%7Ba%5E2%20%7D%20%2B%5Cfrac%7By%5E2%20%7D%7Bb%5E2%7D%20%3D1%EF%BC%88a%EF%BC%9Eb%EF%BC%9E0%EF%BC%89$ $%EF%BC%8CC%E7%9A%84%E4%B8%8A%E9%A1%B6%E7%82%B9%E4%B8%BAA%EF%BC%8C%E4%B8%A4%E4%B8%AA%E7%84%A6%E7%82%B9%E4%B8%BA$ $F_%7B1%7D%EF%BC%8C%20F_%7B2%7D%20%EF%BC%8C%E7%A6%BB%E5%BF%83%E7%8E%87%E4%B8%BA%5Cfrac%7B1%7D%7B2%7D%EF%BC%8C%E8%BF%87F_%7B1%7D%20%20%E4%B8%94%E5%9E%82%E7%9B%B4%E4%BA%8EAF_%7B2%7D%20$ $%E7%9A%84%E7%9B%B4%E7%BA%BF%E4%B8%8EC%E4%BA%A4%E4%BA%8ED%2CE%E4%B8%A4%E7%82%B9%EF%BC%8C%5Cvert%20DE%20%5Cvert$

$%3D6%EF%BC%8C%E5%88%99%CE%94ADE%E7%9A%84%E5%91%A8%E9%95%BF%E4%B8%BA--$

$%E5%9B%A0%E4%B8%BAe%3D%5Cfrac%7Bc%7D%7Ba%7D%3D%5Cfrac%7B1%7D%7B2%7D$

$%E6%89%80%E4%BB%A5a%3D2c%20%EF%BC%8C%20b%3D%5Csqrt%7B3%7D%20c$

$%5Cvert%20OA%20%5Cvert%3D%5Csqrt%7B3%7D%20c%2C%20%5Cvert%20OF_%7B2%7D%20%20%5Cvert%3Dc%20$

$%5Cvert%20AF_%7B1%7D%20%20%5Cvert%3D%20%5Cvert%20AF_%7B2%7D%20%20%5Cvert%20%3D%5Cvert%20F_%7B1%7D%20F_%7B2%7D%20%20%5Cvert%20%3D2c$

$%CE%94AF_%7B1%7D%20F_%7B2%7D%E4%B8%BA%E7%AD%89%E8%BE%B9%E4%B8%89%E8%A7%92%E5%BD%A2%20$

$%E5%9B%A0%E4%B8%BAF_%7B1%7D%20%20%E4%B8%94%E5%9E%82%E7%9B%B4%E4%BA%8EAF_%7B2%7D%20$

$%E6%89%80%E4%BB%A5DE%E5%9E%82%E7%9B%B4%E5%B9%B3%E5%88%86AF_%7B2%7D%20$

$cos%E2%88%A0DF_%7B1%7D%20F_%7B2%7D%20%3D%5Cfrac%7B%5Cpi%20%7D%7B6%7D%20$

$C_%7B%CE%94ADE%7D%3D%20C_%7B%CE%94DEF_%7B2%7D%20%7D%3D4a%20$

$%EF%BC%88%E5%85%A8%E7%AD%89%E4%B8%89%E8%A7%92%E5%BD%A2%EF%BC%8B%E6%A4%AD%E5%9C%86%E5%AE%9A%E4%B9%89%EF%BC%89$

$%E5%9B%A0%E4%B8%BA%5Cvert%20DE%20%5Cvert%20%3D%5Cfrac%7B2ab%5E2%7D%7Ba%5E2-c%5E2cos%5E2%E2%88%A0DF_%7B1%7D%20F_%7B2%7D%20%7D%3D6%20$

$%E5%8D%B3%5Cfrac%7B12c%5E3%7D%7B4c%5E2-%5Cfrac%7B3%7D%7B4%7Dc%5E2%20%7D%3D6%20$

$%E6%89%80%E4%BB%A58c%5E2(c-%5Cfrac%7B13%7D%7B8%7D)%3D0%20$

$c%3D%5Cfrac%7B13%7D%7B8%7D%20$

$C_%7B%CE%94ADE%7D%3D4a%20%3D8c%3D13$

技巧点：

椭圆焦半径公式

$F_%7B1%7D%E4%B8%BA%E5%B7%A6%E7%84%A6%E7%82%B9%EF%BC%8CPQ%E4%B8%BA%E7%84%A6%E7%82%B9%E5%BC%A6%20%EF%BC%8C%5Ctheta%20%E4%B8%BAPQ%E5%80%BE%E6%96%9C%E8%A7%92$

$%5Cvert%20PF_%7B1%7D%20%20%5Cvert%20%3D%5Cfrac%7Bb%5E2%7D%7Ba-ccos%5Ctheta%20%7D%2C%5Cvert%20QF_%7B1%7D%20%20%5Cvert%20%3D%5Cfrac%7Bb%5E2%7D%7Ba%2Bccos%5Ctheta%20%7D%20$