（圓錐曲線）兩點式變換來解決常見問題2

2022-08-25 13:29 作者:因戀相思久 0人讀過 | 我要投稿

已知拋物線 $%5CGamma$ 的頂點在原點，焦點在x軸上，且拋物線上有一點 $P%EF%BC%884%EF%BC%8Ca%EF%BC%89$ 到焦點 $F$ 的距離為5.

（1）求該拋物線 $%5CGamma%20$ 的方程.

（2）已知拋物線上一點 $M%EF%BC%88b%EF%BC%8C4%EF%BC%89$ 過點 $M$ 作拋物線的兩條弦 $MD%EF%BC%8CME$ ，且

$MD%E2%8A%A5ME$ ，判斷直線 $DE$ 是否過定點？并說明理由．

（1）由題可知設(shè)拋物線方程為 $y%5E2%20%3D2px%EF%BC%88p%EF%BC%9E0%EF%BC%89$

所以 $F%EF%BC%88%5Cfrac%7Bp%7D%7B2%7D%20%EF%BC%8C0%EF%BC%89$

由于 $P$ 到焦點 $F$ 的距離為5

所以 $%5Cfrac%7Bp%7D%7B2%7D%20%2B4%3D5$

$p%3D2$

拋物線 $%5CGamma%20%E4%B8%BAy%5E2%20%3D4x$

（2）

直線 $DE$ 斜率存在時

設(shè) $D%EF%BC%88x_%7B1%7D%20%EF%BC%8Cy_%7B1%7D%20%EF%BC%89%EF%BC%8CE%EF%BC%88x_%7B2%7D%20%EF%BC%8Cy_%7B2%7D%20%EF%BC%89$

顯然 $M%EF%BC%884%2C4%EF%BC%89$

直線 $DM%EF%BC%9A%EF%BC%884%2By_%7B1%7D%20%EF%BC%89y%3D4x%2B4y_%7B1%7D%20$

直線 $EM%EF%BC%9A%EF%BC%884%2By_%7B2%7D%20%EF%BC%89y%3D4x%2B4y_%7B2%7D%20$

直線 $DE%EF%BC%9A%EF%BC%88y_%7B1%7D%20%2By_%7B2%7D%20%EF%BC%89y%3D4x%2By_%7B1%7D%20y_%7B2%7D%20$

$%E5%9B%A0%E4%B8%BAMD%E2%8A%A5ME%EF%BC%8C%E6%89%80%E4%BB%A5%5Cfrac%7B4%7D%7B4%2By_%7B1%7D%20%7D%20%5Ccdot%20%5Cfrac%7B4%7D%7B4%2By_%7B2%7D%20%7D%20%3D-1$

整理有

$y_%7B1%7D%20y_%7B2%7D%20%3D-32-4%EF%BC%88y_%7B1%7D%20%2By_%7B2%7D%20%EF%BC%89$

代入直線 $DE$ 并整理

$%EF%BC%88y_%7B1%7D%20%2By_%7B2%7D%20%EF%BC%89%EF%BC%884%2By%EF%BC%89%3D4x-32$

過定點 $%5Cmu%20%EF%BC%888%EF%BC%8C-4%EF%BC%89$

直線 $DE$ 斜率不存在時

不妨設(shè)直線 $DE%E4%B8%BAx%3D8%EF%BC%8CD%EF%BC%888%2C4%5Csqrt%7B2%7D%20%EF%BC%89%EF%BC%8CE%EF%BC%888%EF%BC%8C-4%5Csqrt%7B2%7D%20%EF%BC%89%EF%BC%8CM%EF%BC%884%2C4%EF%BC%89$

$%5Cvec%7BMD%7D%5Ccdot%20%20%5Cvec%7BME%7D%20%3D%EF%BC%884%2C4%5Csqrt%7B2%7D-4%20%EF%BC%89%5Ccdot%20%EF%BC%884%EF%BC%8C-4%5Csqrt%7B2%7D-4%20%EF%BC%89%3D16-32%2B16%3D0$

$%E6%89%80%E4%BB%A5MD%E2%8A%A5ME%EF%BC%8C%E6%BB%A1%E8%B6%B3%E9%A2%98%E6%84%8F$

$%E6%AD%A4%E6%97%B6%E7%9B%B4%E7%BA%BFDE%E8%BF%87%5Cmu%20%EF%BC%888%EF%BC%8C-4%EF%BC%89$

綜上， $%E7%9B%B4%E7%BA%BFDE%E8%BF%87%5Cmu%20%EF%BC%888%EF%BC%8C-4%EF%BC%89$

$%5CDelta%20%5CDelta%20$ 題目改編：

① $%E8%BF%87%E7%82%B9M%E4%BD%9C%E6%8A%9B%E7%89%A9%E7%BA%BF%E7%9A%84%E4%B8%A4%E6%9D%A1%E5%BC%A6MD%EF%BC%8CME$

$k_%7BMD%7D%2B%20k_%7BME%7D%20%3D0%EF%BC%8C%E6%B1%82%E7%9B%B4%E7%BA%BFDE%E7%9A%84%E6%96%9C%E7%8E%87$

② $%E8%BF%87%E7%82%B9M%E4%BD%9C%E6%8A%9B%E7%89%A9%E7%BA%BF%E7%9A%84%E4%B8%A4%E6%9D%A1%E5%BC%A6MD%EF%BC%8CME$

若直線 $MD%EF%BC%8CME$ 與

$%E5%9C%86A%EF%BC%9A%EF%BC%88x-2%EF%BC%89%5E2%2By%5E%202%3D1%E7%9B%B8%E5%88%87$

試探究 $%E5%9C%86A%E4%B8%8E%E7%9B%B4%E7%BA%BFDE%E7%9A%84%E4%BD%8D%E7%BD%AE%E5%85%B3%E7%B3%BB$

① $k_%7BMD%7D%2B%20k_%7BME%7D%20%3D0$ 有

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

整理有 $y_%7B1%7D%2By_%7B2%7D%20%3D-8$

帶入直線DE有

$-8y%3D4x%2By_%7B1%7D%20y_%7B2%7D%20$

斜率為 $-%5Cfrac%7B1%7D%7B2%7D%20$

②直線 $MD%E4%B8%8E%E5%9C%86A%E7%9B%B8%E5%88%87$

由點到直線距離公式有

$%5Cfrac%7B%5Cvert%208%2B4y_%7B1%7D%20%20%5Cvert%20%7D%7B%5Csqrt%7B%EF%BC%884%2By_%7B1%7D%20%EF%BC%89%5E2%20%2B4%5E2%20%7D%20%7D%20%3D1$

整理有

$15y_%7B1%7D%20%5E2%20%2B56y_%7B1%7D%20%2B32%3D0$

同理

$15y_%7B2%7D%20%5E2%20%2B56y_%7B2%7D%20%2B32%3D0$

$%E6%89%80%E4%BB%A5y_%7B1%7D%20y_%7B2%7D%20%3D%5Cfrac%7B32%7D%7B15%7D%20%EF%BC%8Cy_%7B1%7D%20%2By_%7B2%7D%20%3D%5Cfrac%7B-56%7D%7B15%7D%20%EF%BC%88%E5%90%8C%E6%9E%84%2B%E9%9F%A6%E8%BE%BE%EF%BC%89$

$%E8%AE%BE%E7%9B%B4%E7%BA%BFDE%E5%88%B0%E5%9C%86A%E8%B7%9D%E7%A6%BB%E4%B8%BAd$

$d%3D%5Cfrac%7B%5Cvert%208%2By_%7B1%7D%20y_%7B2%7D%20%20%5Cvert%20%20%7D%7B%5Csqrt%7B%EF%BC%88y_%7B2%7D%20%2By_%7B1%7D%20%EF%BC%89%5E2%20%2B4%5E2%20%7D%20%7D%20%3D%5Cfrac%7B%5Cvert%208%2B%5Cfrac%7B32%7D%7B15%7D%5Cvert%20%20%7D%7B%5Csqrt%7B%EF%BC%88%5Cfrac%7B-56%7D%7B15%7D%20%20%EF%BC%89%5E2%20%2B4%5E2%20%7D%20%7D%20%5Capprox%201.85%EF%BC%9E1$