(圆锥曲线)齐次化的应用

2022-07-27 22:17 作者:因恋相思久 0人读过 | 我要投稿

如图

（1）易知a＝2

$%E5%BD%93$ $%E7%82%B9H%E5%9C%A8%E6%A4%AD%E5%9C%86%E4%B8%8A%E6%97%B6$ ，不妨设 $H%EF%BC%881%EF%BC%8C%5Cfrac%7B%5Csqrt%7B3%7D%20b%7D%7B2%7D%20%EF%BC%89$

此时 $R$ = $%5Cvert%20OH%20%5Cvert%20$ = $%5Csqrt%7B1%2B%5Cfrac%7B3b%5E2%20%7D%7B4%7D%20%7D%20$ = $%5Cfrac%7B%5Csqrt%7B13%7D%20%7D%7B2%7D%20$

$%E8%A7%A3%E5%BE%97$ $b%5E2%3D3$

$%E6%A4%AD%E5%9C%86%E6%96%B9%E7%A8%8B%E4%B8%BA$ $%5Cfrac%7Bx%5E2%20%7D%7B4%7D%20%2B%5Cfrac%7By%5E2%20%7D%7B3%7D%20%3D1$

$%EF%BC%882%EF%BC%89$ $%E8%AE%BE%E7%9B%B4%E7%BA%BFPQ%EF%BC%9An%EF%BC%88x%2B2%EF%BC%89%2Bmy%3D1$ ， $P(x_%7B1%7D%2C%20y_%7B1%7D%20)%2CQ(x_%7B2%7D%2C%20y_%7B2%7D%20)$

$%E9%82%A3%E4%B9%88%E6%9C%89k_%7BPQ%7D%20%3D-%5Cfrac%7Bn%7D%7Bm%7D%20$

$%E5%9B%A0%E4%B8%BAk_%7BAQ%7D%20%3Dk_%7BAN%7D%20%3D%5Cfrac%7Bk_%7BON%7D%20%7D%7B3%7D%20$ , $k_%7BAP%7D%20%3Dk_%7BAM%7D%20%3D%5Cfrac%7Bk_%7BOM%7D%20%7D%7B3%7D%20$

$k_%7BAP%7D%20k_%7BAQ%7D%3D%5Cfrac%7Bk_%7BOM%7D%20k_%7BON%7D%7D%7B9%7D%3D%20-%5Cfrac%7B1%7D%7B9%7D%20$ $(OM%5Cbot%20ON%5CRightarrow%20k_%7BOM%7D%20k_%7BON%7D%3D-1%20%20)$

因为H为M,N中点有 $y_%7BOH%7D%20%3D%5Cfrac%7B1%7D%7B2%7D%20%EF%BC%88y_%7BOM%7D%2B%20y_%7BON%7D%20%EF%BC%89$

$%E5%8D%B3k_%7BOH%7D%3D%5Cfrac%7B1%7D%7B2%7D%20%EF%BC%88%20k_%7BON%7D%20%2Bk_%7BOM%7D%20%EF%BC%89$ $%3D%5Cfrac%7B3%7D%7B2%7D%20%EF%BC%88k_%7BAP%7D%2B%20k_%7BAQ%7D%20%EF%BC%89$

$%E4%B8%8B%E9%9D%A2%E5%BC%80%E5%A7%8B%E9%BD%90%E6%AC%A1%E5%8C%96$

$3x%5E2%20%2B4y%5E2%3D12$

$3%EF%BC%88x%2B2-2%EF%BC%89%5E2%20%2B4y%5E2%20%3D12$

$3%EF%BC%88x%2B2%EF%BC%89%5E2-12%EF%BC%88x%2B2%EF%BC%89%20%20%2B4y%5E2%20%3D0$

$3%EF%BC%88x%2B2%EF%BC%89%5E2-12%EF%BC%88x%2B2%EF%BC%89%E3%80%90n%EF%BC%88x%2B2%EF%BC%89%2Bmy%E3%80%91%2B4y%5E2%20%3D0$

$%EF%BC%883-12n%EF%BC%89%EF%BC%88x%2B2%EF%BC%89%5E2-12m%EF%BC%88x%2B2%EF%BC%89y%2B4y%5E2%20%3D0$

$%EF%BC%883-12n%EF%BC%89-12m%EF%BC%88%5Cfrac%7By%7D%7Bx%2B2%7D%20%EF%BC%89%2B4%EF%BC%88%5Cfrac%7By%7D%7Bx%2B2%7D%20%EF%BC%89%5E2%20%3D0$

故 $k_%7BAP%7D%EF%BC%8C%20k_%7BAQ%7D%20%E4%B8%BA%E6%96%B9%E7%A8%8B%EF%BC%883-12n%EF%BC%89-12mk%2B4k%5E2%20%3D0%E7%9A%84%E4%B8%A4%E6%A0%B9$

$k_%7BAP%7D%20k_%7BAQ%7D$ = $%5Cfrac%7B3-12n%7D%7B4%7D%20$ $k_%7BAP%7D%2B%20k_%7BAQ%7D%20%3D3m$

所以 $k_%7BOH%7D%3D%20%5Cfrac%7B9%7D%7B2%7D%20m$ ， $k_%7BAP%7D%20k_%7BAQ%7D$ = $%5Cfrac%7B3-12n%7D%7B4%7D%20$ = $-%5Cfrac%7B1%7D%7B9%7D%20$ $%5CRightarrow%20%5Cfrac%7B9%7D%7B2%7D%20n%3D$ $%5Cfrac%7B31%7D%7B24%7D%20$

$k_%7B1%7D%20k_%7B2%7D%20%3Dk_%7BOH%7D%20k_%7BPQ%7D%20%3D-%5Cfrac%7Bn%7D%7Bm%7D%20%5Ctimes%20$ $%5Cfrac%7B9m%7D%7B2%7D%20$ = $-%5Cfrac%7B31%7D%7B24%7D%20$ 为定值

$%E5%8F%A6%E5%A4%96%E5%8F%91%E7%8E%B0%E6%8A%8A%E6%B1%82%E5%87%BAn%3D%5Cfrac%7B31%7D%7B108%7D%20%E4%BB%A3%E8%BF%9B%E7%9B%B4%E7%BA%BFPQ%0A$ $%E6%9C%89$

$%5Cfrac%7B31%7D%7B108%7D%20%EF%BC%88x-%5Cfrac%7B46%7D%7B31%7D%20%EF%BC%89%2Bmy%3D0$

$%E8%BF%87%E5%AE%9A%E7%82%B9%5Cvarphi%20%EF%BC%88%5Cfrac%7B46%7D%7B31%7D%2C0%20%EF%BC%89$

$%E6%80%BB%E7%BB%93$

关于圆：圆心Q一般有直径B,C与圆上一点A，一般有

$%5Cvec%7BAB%7D%5Ccdot%20%20%5Cvec%7BAC%7D%20%3D0$ ， $%5Cvec%7BAQ%7D%3D%20%5Cfrac%7B1%7D%7B2%7D%20%EF%BC%88%5Cvec%7BAB%7D%2B%20%5Cvec%7BAC%7D%20%EF%BC%89$