抛物线过定点问题（两点式解法）

2023-02-03 16:51 作者:因恋相思久 0人读过 | 我要投稿

17.过抛物线 $C%EF%BC%9Ay%5E2%20%3D4x$ 的焦点 $F$ 的直线 $l$ 交抛物线 $C$ 于 $A%EF%BC%8CB$ 两点，若 $A$ 关于 $x$ 轴的对称点为 $D$ ，求证：直线 $BD$ 过定点，并求出该定点

$%E8%A7%A3%EF%BC%9A%E8%AE%BEA%EF%BC%88x_%7B1%7D%20%EF%BC%8Cy_%7B1%7D%20%EF%BC%89%EF%BC%8CB%EF%BC%88x_%7B2%7D%20%EF%BC%8Cy_%7B2%7D%20%EF%BC%89%2CD%EF%BC%88x_%7B1%7D%20%EF%BC%8C-y_%7B1%7D%20%EF%BC%89$

$%EF%BC%88%E7%9B%B4%E7%BA%BF%E6%96%9C%E7%8E%87%E5%AD%98%E5%9C%A8%EF%BC%89$

$%E7%9B%B4%E7%BA%BFAB%EF%BC%9A%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%EF%BC%89y%3D4x%2B%20y_%7B1%7D%20y_%7B2%7D%20$

$%E7%9B%B4%E7%BA%BFBD%EF%BC%9A%EF%BC%88-y_%7B1%7D%2B%20y_%7B2%7D%EF%BC%89y%3D4x-%20y_%7B1%7D%20y_%7B2%7D%20$

$%E7%9B%B4%E7%BA%BFAB%E8%BF%87F%EF%BC%881%EF%BC%8C0%EF%BC%89$

$%E6%89%80%E4%BB%A5y_%7B1%7D%20y_%7B2%7D%20%3D-4$

$%E6%89%80%E4%BB%A5%E7%9B%B4%E7%BA%BFBD%EF%BC%9A%EF%BC%88-y_%7B1%7D%2B%20y_%7B2%7D%EF%BC%89y%3D4x%2B4$

$%E8%BF%87%E5%AE%9A%E7%82%B9%EF%BC%88-1%EF%BC%8C0%EF%BC%89$

18.已知抛物线 $C%EF%BC%9Ay%5E2%20%3D4x$ ， $A%EF%BC%881%EF%BC%8C2%EF%BC%89$ 过 $A$ 作 $C$ 的两条动弦 $AD%EF%BC%8CAE$ ，且 $k_%7BAD%7D%20k_%7BAE%7D%20%3D2$ ，直线 $DE$ 是否过定点？若是，求出定点坐标，若否，说明理由

$%E8%A7%A3%EF%BC%9A%E8%AE%BED%EF%BC%88x_%7B1%7D%20%EF%BC%8Cy_%7B1%7D%20%EF%BC%89%EF%BC%8CE%EF%BC%88x_%7B2%7D%20%EF%BC%8Cy_%7B2%7D%20%EF%BC%89$

（直线斜率存在）

$%E7%9B%B4%E7%BA%BFAD%EF%BC%9A%EF%BC%88y_%7B1%7D%2B%202%EF%BC%89y%3D4%2B%202y_%7B1%7D%20%20$

$%E7%9B%B4%E7%BA%BFAE%EF%BC%9A%EF%BC%88y_%7B2%7D%2B%202%EF%BC%89y%3D4%2B2%20y_%7B2%7D%20$

$%E7%9B%B4%E7%BA%BFDE%EF%BC%9A%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%20%EF%BC%89y%3D4x%2B%20y_%7B1%7D%20%20y_%7B2%7D%20%5Cotimes%20$

$%E5%9B%A0%E4%B8%BAk_%7BAD%7D%20k_%7BAE%7D%20%3D2$

$%E6%89%80%E4%BB%A5%5Cfrac%7B16%7D%7B%EF%BC%88y_%7B1%7D%20%2B2%EF%BC%89%EF%BC%88y_%7B2%7D%20%2B2%EF%BC%89%7D%20%3D2$

$%E5%8D%B3-2%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%20%EF%BC%89%3D-4%2B%20y_%7B1%7D%20%20y_%7B2%7D%20%5Coplus%20$

$%5Cotimes%20-%5Coplus%20%E5%BE%97$

$%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%20%EF%BC%89%EF%BC%88y%2B2%EF%BC%89%3D4x%2B%204$

$%E8%BF%87%E5%AE%9A%E7%82%B9%EF%BC%88-1%EF%BC%8C-2%EF%BC%89$

19.已知抛物线 $E%EF%BC%9Ay%5E2%20%3D4x$ , $M%2CN$ 为直线 $x%3D-2$ 上的两点， $M%EF%BC%8CN$ 两点的纵坐标之积为 $-8$ ， $P$ 为抛物线上一动点， $PM%EF%BC%8CPN%EF%BC%8C$ 分别交抛物线于 $A%EF%BC%8CB$ 两点，问：直线 $AB$ 是否过定点，若是，求出该定点，若否，说明理由

$%E8%A7%A3%EF%BC%9A%E8%AE%BEA%EF%BC%88x_%7B1%7D%20%EF%BC%8Cy_%7B1%7D%20%EF%BC%89%EF%BC%8CB%EF%BC%88x_%7B2%7D%20%EF%BC%8Cy_%7B2%7D%20%EF%BC%89%EF%BC%8CP%EF%BC%88x_%7B0%7D%20%EF%BC%8Cy_%7B0%7D%20%EF%BC%89$

$M%EF%BC%88-2%EF%BC%8Cy_%7BM%7D%20%EF%BC%89%EF%BC%8CN%EF%BC%88-2%20%EF%BC%8Cy_%7BN%7D%20%EF%BC%89$

（直线斜率存在）

$%E7%9B%B4%E7%BA%BFAP%EF%BC%9A%EF%BC%88y_%7B1%7D%2B%20y_%7B0%7D%EF%BC%89y%3D4x%2B%20y_%7B1%7D%20y_%7B0%7D%20$

$%E7%9B%B4%E7%BA%BFBP%EF%BC%9A%EF%BC%88y_%7B2%7D%2B%20y_%7B0%7D%EF%BC%89y%3D4x%2B%20y_%7B2%7D%20y_%7B0%7D%20$

$%E7%9B%B4%E7%BA%BFAB%EF%BC%9A%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%EF%BC%89y%3D4x%2B%20y_%7B1%7D%20y_%7B2%7D%20$

$%E7%9B%B4%E7%BA%BFAP%E8%BF%87M%EF%BC%8C%E7%9B%B4%E7%BA%BFBP%E8%BF%87N$

$%E5%9B%A0%E6%AD%A4%EF%BC%88y_%7B1%7D%2B%20y_%7B0%7D%EF%BC%89y_%7BM%7D%20%3D-8%2B%20y_%7B1%7D%20y_%7B0%7D%20%5Cotimes%20$

$%EF%BC%88y_%7B2%7D%2B%20y_%7B0%7D%EF%BC%89y_%7BN%7D%20%3D-8%2B%20y_%7B2%7D%20y_%7B0%7D%20%5Coplus%20$

$%5Cotimes%20%5Cast%20%5Coplus%20%E5%BE%97$

$%E6%89%80%E4%BB%A5(y_%7B1%7D%2B%20y_%7B0%7D)(y_%7B2%7D%2B%20y_%7B0%7D)y_%7BM%7Dy_%7BN%7D%20%20%3D(-8%2B%20y_%7B1%7D%20y_%7B0%7D%20)(-8%2By_%7B2%7D%20y_%7B0%7D%20)$

$%E5%9B%A0%E4%B8%BAy_%7BM%7D%20y_%7BN%7D%20%3D-8%2C%E4%BB%A3%E5%85%A5%E5%BE%97$

$-8%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%EF%BC%89y_%7B0%7D%20-8%20y_%7B1%7D%20y_%7B2%7D%20-8y_%7B0%7D%5E2%3D%20-8%EF%BC%88y_%7B1%7D%2B%20y_%7B2%7D%EF%BC%89y_%7B0%7D%20%2B64%2B%20y_%7B1%7D%20y_%7B2%7Dy_%7B0%7D%5E2$