圓錐曲線之流程思路

2022-08-31 18:23 作者:因戀相思久 0人讀過 | 我要投稿

$21.%E5%B7%B2%E7%9F%A5%E6%8A%9B%E7%89%A9%E7%BA%BFC%EF%BC%9Ay%5E2%20%3D2px%EF%BC%88p%EF%BC%9E0%EF%BC%89$ 的焦點為 $F$ ，在曲線 $C$ 上存在一點 $Q%EF%BC%88x_%7B0%7D%EF%BC%8Cp%20%EF%BC%89$ 滿足 $%5Cvert%20QF%20%5Cvert%20%3D2$ ，過原點作兩條相互垂直的直線交曲線 $C$ 于異于原點的兩點 $A%EF%BC%8CB.$

（1）求證：直線 $AB$ 與 $x$ 軸相交于定點 $N$

（2）試探究 $x$ 軸上是否存在定點 $M$ 滿足 $%5Cfrac%7BS_%7B%E2%96%B3ANM%20%7D%20%7D%7BS_%7B%E2%96%B3BNM%7D%20%7D%3D%20%5Cfrac%7BAM%7D%7BBM%7D%20$ 恒成立，若存在，請求出 $M$ 點坐標，若不存在，請說明理由.

（1）①求方程：知道方程才能解決一切問題

$%EF%BC%88%E7%84%A6%E7%82%B9F%EF%BC%8CQ%E5%9C%A8C%E4%B8%8A%EF%BC%8C%5Cvert%20QF%20%5Cvert%3D2%20%EF%BC%89%5CRightarrow%20%E6%8A%9B%E7%89%A9%E7%BA%BF$

$%E5%AE%9A%E4%B9%89%5CRightarrow%20%E5%88%97%E5%85%B3%E4%BA%8Ep%E6%96%B9%E7%A8%8B%5CRightarrow%20%E6%8A%9B%E7%89%A9%E7%BA%BFC$

②看問題： $%E7%9B%B4%E7%BA%BFAB%E4%B8%8Ex%E8%BD%B4%E4%B8%BA%E5%AE%9A%E7%82%B9N$

$%E8%A7%82%E5%AF%9F%E7%9B%B4%E7%BA%BF%E4%B8%ADA%EF%BC%8CB%5CRightarrow%20%EF%BC%88%E5%8E%9F%E7%82%B9%EF%BC%8C%E5%9E%82%E7%9B%B4%EF%BC%89$

$%5CRightarrow%20%20k_%7BOA%7D%20k_%7BOB%7D%E4%B8%BA%E5%AE%9A%E5%80%BC%20%5CRightarrow%20%E5%AE%9A%E7%82%B9N$

③求定點

$%E4%B8%A4%E7%82%B9%E5%BC%8F%E5%8F%98%E6%8D%A2%EF%BC%8C%E9%BD%90%E6%AC%A1%E5%8C%96%EF%BC%8C%E9%9F%A6%E8%BE%BE%E7%AD%89%5CRightarrow%20%E7%BB%93%E5%90%88$

$k_%7BOA%7D%20k_%7BOB%7D%20%5CRightarrow%20%E7%9B%B4%E7%BA%BFAB%5CRightarrow%20%E5%AE%9A%E7%82%B9N$

解： $%E5%9B%A0%E4%B8%BA%E7%84%A6%E7%82%B9P%EF%BC%88%5Cfrac%7Bp%7D%7B2%7D%20%2C0%EF%BC%89%EF%BC%8CQ%EF%BC%88x_%7B0%7D%EF%BC%8Cp%20%EF%BC%89$

根據(jù)拋物線定義有

$x_%7B0%7D%20%2B%5Cfrac%7Bp%7D%7B2%7D%20%3D2%EF%BC%8Cp%5E2%3D2px_%7B0%7D%20%20$

$%E8%A7%A3%E5%BE%97p%3D2%EF%BC%8Cx_%7B0%7D%20%3D1$

拋物線方程為 $y%5E2%20%3D4x$

設(shè) $A%EF%BC%88%5Cfrac%7By_%7B1%7D%5E2%20%7D%7B4%7D%20%EF%BC%8Cy_%7B1%7D%EF%BC%89B%EF%BC%88%5Cfrac%7By_%7B2%7D%20%5E2%7D%7B4%7D%EF%BC%8Cy_%7B2%7D%20%EF%BC%89$

$%E9%82%A3%E4%B9%88%5Cfrac%7By_%7B1%7D%20%7D%7Bx_%7B1%7D%20%7D%20%3D%5Cfrac%7B4%7D%7By_%7B1%7D%20%7D%20%EF%BC%8C%5Cfrac%7By_%7B2%7D%20%7D%7Bx_%7B2%7D%20%7D%20%3D%5Cfrac%7B4%7D%7By_%7B2%7D%20%7D%20$

$%E6%89%80%E4%BB%A5k_%7BOA%7D%20k_%7BOB%7D%20%3D%5Cfrac%7B16%7D%7By_%7B1%7D%20y_%7B2%7D%20%7D%20%3D-1$

$%E5%8D%B3y_%7B1%7D%20y_%7B2%7D%20%3D-16$

$%E7%9B%B4%E7%BA%BFAB%EF%BC%9Ay-y_%7B1%7D%20%3D%5Cfrac%7By_%7B1%7D%20-y_%7B2%7D%20%7D%7B%5Cfrac%7By_%7B1%7D%20%5E2%7D%7B4%7D-%20%5Cfrac%7By_%7B2%7D%20%5E2%7D%7B4%7D%20%7D%20%EF%BC%88x-%5Cfrac%7By_%7B1%7D%5E2%20%7D%7B4%7D%20%EF%BC%89$

$%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%20%EF%BC%89y%3D4x%2By_%7B1%7D%20y_%7B2%7D$

$%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%20%EF%BC%89%3D4x-16$

$%E8%BF%87%E5%AE%9A%E7%82%B9N%EF%BC%884%2C0%EF%BC%89$

（2）①看條件：充分利用小問給出的條件

$%EF%BC%88%5Cfrac%7BS_%7B%E2%96%B3ANM%20%7D%20%7D%7BS_%7B%E2%96%B3BNM%7D%20%7D%3D%20%5Cfrac%7BAM%7D%7BBM%7D%20%EF%BC%89%5CRightarrow%20%5Cfrac%7B%5Cfrac%7B1%7D%7B2%7D%20AM%5Ccdot%20h_%7B1%7D%7D%7B%5Cfrac%7B1%7D%7B2%7D%20BM%5Ccdot%20h_%7B2%7D%7D%20%3D%5Cfrac%7BAM%7D%7BBM%7D%20$

（左右盡量找出關(guān)系是很重要的）

$%5CRightarrow%20h_%7B1%7D%3D%20h_%7B2%7D%20%5CRightarrow%20N%E5%88%B0%E7%9B%B4%E7%BA%BFAM%EF%BC%8CAN%E8%B7%9D%E7%A6%BB%E7%9B%B8%E5%90%8C$

$%5CRightarrow%20%E2%88%A0AMN%3D%E2%88%A0BMN%EF%BC%88%E8%A7%92%E5%B9%B3%E5%88%86%E7%BA%BF%EF%BC%89%5CRightarrow%20$

$tan%5Calpha%20%2Btan%5Cbeta%20%3D0%5CRightarrow%20k_%7BAM%7D%2B%20k_%7BBM%7D%20%3D0%5CRightarrow%20$

$%E5%88%97%E5%BC%8F%5CRightarrow%20y_%7B1%7D%20y_%7B2%7D%20%EF%BC%8Cy_%7B1%7D%2B%20y_%7B2%7D%5CRightarrow%20%E9%9F%A6%E8%BE%BE%5CRightarrow%20M$

解：

$%E6%96%9C%E7%8E%87%E5%AD%98%E5%9C%A8%E6%97%B6%EF%BC%8C%E8%AE%BE%E7%9B%B4%E7%BA%BFAB%EF%BC%9Any%2B4%3Dx$

$%E8%AE%BE%E7%9B%B4%E7%BA%BFAM%EF%BC%8CBM%E5%88%B0N%E7%9A%84%E8%B7%9D%E7%A6%BB%E4%B8%BA%E5%88%86%E5%88%AB%E4%B8%BAd_%7B1%7D%20%EF%BC%8C%2Cd_%7B2%7D%20$

$%E5%9B%A0%E4%B8%BA%5Cfrac%7BS_%7B%E2%96%B3ANM%20%7D%20%7D%7BS_%7B%E2%96%B3BNM%7D%20%7D%3D%20%5Cfrac%7BAM%7D%7BBM%7D$

$%E6%89%80%E4%BB%A5%5Cfrac%7B%5Cfrac%7B1%7D%7B2%7D%20AM%5Ccdot%20d_%7B1%7D%7D%7B%5Cfrac%7B1%7D%7B2%7D%20BM%5Ccdot%20d_%7B2%7D%7D%3D%5Cfrac%7BAM%7D%7BBM%7D%20$

$%E5%8D%B3d_%7B1%7D%20%3Dd_%7B2%7D%20$

$%E6%89%80%E4%BB%A5%E2%88%A0AMN%3D%E2%88%A0BMN$

$%E6%89%80%E4%BB%A5%20k_%7BAM%7D%2B%20k_%7BBM%7D%20%3D0$

$%E5%8D%B3%5Cfrac%7By_%7B1%7D%20%7D%7Bx_%7B1%7D-t%20%7D%20%2B%5Cfrac%7By_%7B2%7D%20%7D%7Bx_%7B2%7D-t%20%7D%20%3D0$

$%5Cfrac%7By_%7B1%7D%20%7D%7Bny_%7B1%7D%2B4-t%20%7D%20%2B%5Cfrac%7By_%7B2%7D%20%7D%7Bny_%7B2%7D%2B4-t%20%7D%20%3D0$

$2ny_%7B1%7D%20y_%7B2%7D%20%2B%EF%BC%884-t%EF%BC%89%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%20%EF%BC%89%3D0$

$%E7%9B%B4%E7%BA%BFAB%EF%BC%9Any%2B4%3Dx%E4%B8%8Ey%5E2%3D4x%E8%81%94%E7%AB%8B$

$y%5E2-4n%20y-16%3D0$

$y_%7B1%7D%20y_%7B2%7D%20%3D-16%EF%BC%8Cy_%7B1%7D%20%2By_%7B2%7D%3D4n$

$%E6%89%80%E4%BB%A5-32%2B4%EF%BC%884-t%EF%BC%89%3D0$

$%E8%A7%A3%E5%BE%97t%3D-4$

$%E6%89%80%E4%BB%A5N%EF%BC%88-4%2C0%EF%BC%89$

$%E6%96%9C%E7%8E%87%E4%B8%8D%E5%AD%98%E5%9C%A8%E6%97%B6%EF%BC%8CN%EF%BC%88-4%2C0%EF%BC%89%E4%B9%9F%E6%BB%A1%E8%B6%B3%E9%A2%98%E6%84%8F$

綜上

$N%EF%BC%88-4%2C0%EF%BC%89$

$%E5%AF%B9%E4%BA%8E%5Cfrac%7By_%7B1%7D%20%7D%7Bx_%7B1%7D-t%20%7D%20%2B%5Cfrac%7By_%7B2%7D%20%7D%7Bx_%7B2%7D-t%20%7D%20%3D0$

不用韋達也可以做

$%E5%8C%96%E7%AE%80%E6%9C%89x_%7B1%7D%20y_%7B2%7D%2B%20x_%7B2%7D%20y_%7B1%7D%20%3Dt%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%20%EF%BC%89$

$%E5%8D%B3%5Cfrac%7By_%7B1%7D%5E2%20%7D%7B%204%20%7D%20%20y_%7B2%7D%2B%20%5Cfrac%7By_%7B2%7D%5E2%20%7D%7B4%7D%20%20y_%7B1%7D%20%3Dt%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%20%EF%BC%89$