设A(-C,0) B(C,0) (C>0)为两定点,东、动点P到A带内的距离与到B点的距离的比值为定值小A(小A>0),求P点的轨迹
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/01 23:24:56
![设A(-C,0) B(C,0) (C>0)为两定点,东、动点P到A带内的距离与到B点的距离的比值为定值小A(小A>0),求P点的轨迹](/uploads/image/z/3713698-10-8.jpg?t=%E8%AE%BEA%EF%BC%88-C%2C0%EF%BC%89+B%EF%BC%88C%2C0%EF%BC%89+%EF%BC%88C%3E0%EF%BC%89%E4%B8%BA%E4%B8%A4%E5%AE%9A%E7%82%B9%2C%E4%B8%9C%E3%80%81%E5%8A%A8%E7%82%B9P%E5%88%B0A%E5%B8%A6%E5%86%85%E7%9A%84%E8%B7%9D%E7%A6%BB%E4%B8%8E%E5%88%B0B%E7%82%B9%E7%9A%84%E8%B7%9D%E7%A6%BB%E7%9A%84%E6%AF%94%E5%80%BC%E4%B8%BA%E5%AE%9A%E5%80%BC%E5%B0%8FA%EF%BC%88%E5%B0%8FA%3E0%EF%BC%89%2C%E6%B1%82P%E7%82%B9%E7%9A%84%E8%BD%A8%E8%BF%B9)
设A(-C,0) B(C,0) (C>0)为两定点,东、动点P到A带内的距离与到B点的距离的比值为定值小A(小A>0),求P点的轨迹
设A(-C,0) B(C,0) (C>0)为两定点,东、动点P到A带内的距离与到B点的距离的比值为定值小A(小A>0),求P点的轨迹
设A(-C,0) B(C,0) (C>0)为两定点,东、动点P到A带内的距离与到B点的距离的比值为定值小A(小A>0),求P点的轨迹
解;设P(x,y)则有
√(x+c)^2+y^2/√(x-C)^2+y^2=a
(x+c)^2+y^2=a^2(x-C)^2+a^2y^2
即x^2+2cx+c^2+y^2=a^2(x^2-2cx+c^2)+a^2y^2
(a^2-1)x^2-2c(a^2+1)x+(a^2-1)c^2+(a^2-1)y^2=0
(1)当a=1时,x=0,P的轨迹为AB的中垂线
(2)a≠0
(a^2-1)(x-c(a^2+1)/(a^2-1))^2+(a^2-1)y^2
=c^2(a^2+1)^2/(a^2-1)-(a^2-1)c^2=c^2*4a^2/(a^2-1)
(x-c(a^2+1)/(a^2-1))^2+y^2=4a^2c^2/(a^2-1)^2
则P的轨迹为圆,圆心在(c(a^2+1)/(a^2-1),0),半径为2ac/|a^2-1|
设A(-C,0) B(C,0) (C>0)为两定点,东、动点P到A带内的距离与到B点的距离的比值为定值小A(小A>0),求P点的轨迹
解;设P(x,y)则有
√[(x+c)^2+y^2]/√[(x-C)^2+y^2]=a
(x+c)^2+y^2=a^2(x-C)^2+a^2y^2
即x^2+2cx+c^2+y^2=a^2(x^2-2cx+c^2)+a^2y^2
(a^2-1)x^2-2c(a^2+1)x+(a^2-1)c^2+(a^2-1)y^2=0
(1)当a=1时,P点的轨迹方程为x=...
全部展开
解;设P(x,y)则有
√[(x+c)^2+y^2]/√[(x-C)^2+y^2]=a
(x+c)^2+y^2=a^2(x-C)^2+a^2y^2
即x^2+2cx+c^2+y^2=a^2(x^2-2cx+c^2)+a^2y^2
(a^2-1)x^2-2c(a^2+1)x+(a^2-1)c^2+(a^2-1)y^2=0
(1)当a=1时,P点的轨迹方程为x=0,P的轨迹为AB的中垂线即y轴;
(2)当a=0时,P点的轨迹为点A(-c,0);
(3)a≠0,且a≠1时,
(a^2-1)(x-c(a^2+1)/(a^2-1))^2+(a^2-1)y^2
=c^2(a^2+1)^2/(a^2-1)-(a^2-1)c^2=c^2*4a^2/(a^2-1)^2
(x-c(a^2+1)/(a^2-1))^2+y^2=4a^2c^2/(a^2-1)^2
则P的轨迹为圆,圆心在(c(a^2+1)/(a^2-1),0),半径为2ac/|a^2-1| .
收起