已知F1(-c,O),F2(c,0)为椭圆x²/a²+y²/b²=1(a>b>0)的两个焦点.已知F1(-c,O),F2(c,0)为椭圆x²/a²+y²/b²=1(a>b>0)的两个焦点,P为椭圆上一点且向量PF1·向量PF2=c²,则此椭圆离心率的
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/20 05:53:16
![已知F1(-c,O),F2(c,0)为椭圆x²/a²+y²/b²=1(a>b>0)的两个焦点.已知F1(-c,O),F2(c,0)为椭圆x²/a²+y²/b²=1(a>b>0)的两个焦点,P为椭圆上一点且向量PF1·向量PF2=c²,则此椭圆离心率的](/uploads/image/z/3704111-71-1.jpg?t=%E5%B7%B2%E7%9F%A5F1%28-c%2CO%29%2CF2%28c%2C0%29%E4%B8%BA%E6%A4%AD%E5%9C%86x%26%23178%3B%2Fa%26%23178%3B%2By%26%23178%3B%2Fb%26%23178%3B%3D1%28a%3Eb%3E0%29%E7%9A%84%E4%B8%A4%E4%B8%AA%E7%84%A6%E7%82%B9.%E5%B7%B2%E7%9F%A5F1%28-c%2CO%29%2CF2%28c%2C0%29%E4%B8%BA%E6%A4%AD%E5%9C%86x%26%23178%3B%2Fa%26%23178%3B%2By%26%23178%3B%2Fb%26%23178%3B%3D1%28a%3Eb%3E0%29%E7%9A%84%E4%B8%A4%E4%B8%AA%E7%84%A6%E7%82%B9%2CP%E4%B8%BA%E6%A4%AD%E5%9C%86%E4%B8%8A%E4%B8%80%E7%82%B9%E4%B8%94%E5%90%91%E9%87%8FPF1%C2%B7%E5%90%91%E9%87%8FPF2%3Dc%26%23178%3B%2C%E5%88%99%E6%AD%A4%E6%A4%AD%E5%9C%86%E7%A6%BB%E5%BF%83%E7%8E%87%E7%9A%84)
已知F1(-c,O),F2(c,0)为椭圆x²/a²+y²/b²=1(a>b>0)的两个焦点.已知F1(-c,O),F2(c,0)为椭圆x²/a²+y²/b²=1(a>b>0)的两个焦点,P为椭圆上一点且向量PF1·向量PF2=c²,则此椭圆离心率的
已知F1(-c,O),F2(c,0)为椭圆x²/a²+y²/b²=1(a>b>0)的两个焦点.
已知F1(-c,O),F2(c,0)为椭圆x²/a²+y²/b²=1(a>b>0)的两个焦点,P为椭圆上一点且向量PF1·向量PF2=c²,则此椭圆离心率的取值范围是_____.
已知F1(-c,O),F2(c,0)为椭圆x²/a²+y²/b²=1(a>b>0)的两个焦点.已知F1(-c,O),F2(c,0)为椭圆x²/a²+y²/b²=1(a>b>0)的两个焦点,P为椭圆上一点且向量PF1·向量PF2=c²,则此椭圆离心率的
向量PF1•向量PF2=|PF1|*|PF2|cos∠F1PF2=c²
设坐标 P(x,y),则向量 PF1={-c-x,-y},向量 PF2={c-x,-y};
向量PF1•向量PF2=-(c-x)²+y² =c²;
∴ y²=c²+(c-x)²=2c²-2cx+x²,代入椭圆方程:x²/a²+[2c²-2cx+x²]/b²=1;
将方程通分化简 (a²+b²)x²-2a²cx+a²(2c²-b²)=0;
当 △=(2a²c)²-4(a²+b²)*a²(2c²-b²)≥0 时方程有解(即 P 才存在),∴ a²c²-(2a²-c²)(3c²-a²)≥0;
展开并以 e=c/a 代入:3(e²)²-6e²+2≥0,∴ e²≤1-√3/3;
0