湊！狠狠的湊！2023甲卷圓曲個人較爛的寫法

2023-07-30 16:01 作者:因戀相思久 0人讀過 | 我要投稿

20.已知直線 $x-2y%2B1%3D0$ 與拋物線 $C%3Ay%5E2%3D2px(p%3E0)%20$ 交于 $A%EF%BC%8CB$ 兩點，且 $%5Cvert%20AB%5Cvert%3D4%5Csqrt%7B15%7D%20%20$

$(1)%E6%B1%82p$

$(2)%E8%AE%BEC%E7%9A%84%E7%84%A6%E7%82%B9%E4%B8%BAF%EF%BC%8CM%2CN%E4%B8%BAC%E4%B8%8A%E4%B8%A4%E7%82%B9$ $%5Cvec%7BMF%7D%20%C2%B7%5Cvec%7BNF%7D%20%3D0%EF%BC%8C%E6%B1%82%5CDelta%20MNF%E9%9D%A2%E7%A7%AF%E7%9A%84%E6%9C%80%E5%B0%8F%E5%80%BC$

$(1).%E8%AE%BEA%EF%BC%88x_%7B1%7D%20%EF%BC%8Cy_%7B1%7D%20%EF%BC%89%EF%BC%8CB%EF%BC%88x_%7B2%7D%20%EF%BC%8Cy_%7B2%7D%20%EF%BC%89$

$%E7%9B%B4%E7%BA%BFAB%E4%B8%8EC%E8%81%94%E7%AB%8B%E6%9C%89$

$y%5E2-4py-2p%3D0%20%5CRightarrow%20y_%7B1%7D%20%2B%20y_%7B2%7D%3D4p%2C%20y_%7B1%7D%20y_%7B2%7D%3D2p$

$%5Cvert%20AB%5Cvert%3D%5Csqrt%7B1%2B%5Cfrac%7B1%7D%7Bk%5E2%7D%20%20%7D%20%5Csqrt%7B(y_%7B1%7D%20%2B%20y_%7B2%7D)%5E2-4y_%7B1%7D%20y_%7B2%7D%7D%3D%204%5Csqrt%7B15%7D%20$

$%E5%8D%B3%5Csqrt%7B5%7D%20%5Csqrt%7B16p%5E2%20-8p%7D%20%3D4%5Csqrt%7B15%7D%20%5CRightarrow%202p%5E2-p-6%3D0$

$%E8%A7%A3%E5%BE%97p%3D2%E6%89%80%E4%BB%A5C%3Ay%5E2%3D4x%20$

$(2).M%EF%BC%88x_%7B3%7D%20%EF%BC%8Cy_%7B3%7D%20%EF%BC%89%EF%BC%8CN%EF%BC%88x_%7B4%7D%20%EF%BC%8Cy_%7B4%7D%20%EF%BC%89%E7%9B%B4%E7%BA%BFMN%EF%BC%9An%EF%BC%88x-1%EF%BC%89%2Bmy%3D1$

$%E7%9B%B4%E7%BA%BFMN%E4%B8%8EC%E8%81%94%E7%AB%8B%E6%9C%89$

$%EF%BC%884m%5E2-1%EF%BC%89(%5Cfrac%7By%7D%7Bx-1%7D%20)%5E2%2B(4m%2B8nm)(%5Cfrac%7By%7D%7Bx-1%7D%20)%2B4n%5E2%2B4n%3D0$

$%5Cvec%7BMF%7D%20%C2%B7%5Cvec%7BNF%7D%20%3D0%5CRightarrow%20k_%7B3%7D%20k_%7B4%7D%3D-1%20%5CRightarrow%20%20%5Cfrac%7By_%7B3%7D%20%7D%7Bx_%7B3%7D-1%20%7D%20%5Cfrac%7By_%7B4%7D%20%7D%7Bx_%7B4%7D-1%20%7D%3D-1%20$

$%E6%89%80%E4%BB%A5%20%5Cfrac%7By_%7B3%7D%20%7D%7Bx_%7B3%7D-1%20%7D%20%5Cfrac%7By_%7B4%7D%20%7D%7Bx_%7B4%7D-1%20%7D%3D%5Cfrac%7B4n%5E2%2B4n%7D%7B4m%5E2-1%7D%3D%20-1%20$

$%5CRightarrow%204m%5E2%3D1-4n-4n%5E2%3E0$

$n%5Cin%20(%5Cfrac%7B-1-%5Csqrt%7B2%7D%20%7D%7B2%7D%20%2C%5Cfrac%7B-1%2B%5Csqrt%7B2%7D%20%7D%7B2%7D)$

$S_%7B%5CDelta%20MNF%20%7D%3D%5Cfrac%7B1%7D%7B2%7D%20%20%5Cvert%20NF%20%5Cvert%20%5Cvert%20MF%20%5Cvert%20%3D%5Cfrac%7B1%7D%7B2%7D%20(x_%7B3%7D%2B1%20)(x_%7B4%7D%2B1%20)$

（拋物線定義）

$%E7%9B%B4%E7%BA%BFMN%E4%B8%8EC%E8%81%94%E7%AB%8B%E6%9C%89$

$n%5E2(x%2B1)%5E%202-%5B2n(1%2B2n)%20%2B4m%5E2%5D(x%2B1)%2B(1%2B2n)%5E2%2B4m%5E2%3D0%20$

$(x_%7B3%7D%2B1%20)(x_%7B4%7D%2B1%20)%3D%5Cfrac%7B4m%5E2%2B(1%2B2n)%5E2%7D%7Bn%5E2%7D%20$

$%E6%95%85S%3D%5Cfrac%7B1%7D%7B2%7D%20(x_%7B3%7D%2B1%20)(x_%7B4%7D%2B1%20)%3D%5Cfrac%7B1-4n-4n%5E2%2B1%2B4n%2B4n%5E2%7D%7B2n%5E2%7D%3D%5Cfrac%7B1%7D%7Bn%5E2%7D%20%20$