设椭圆的方程为X平方+Y平方/4=1,过M(0,1)的直线交椭圆于AB两点,O为坐标原点,OP向量=1/2(OA向量+OB向量),当L绕点M旋转时,求动点P的轨迹方程
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![设椭圆的方程为X平方+Y平方/4=1,过M(0,1)的直线交椭圆于AB两点,O为坐标原点,OP向量=1/2(OA向量+OB向量),当L绕点M旋转时,求动点P的轨迹方程](/uploads/image/z/3801867-51-7.jpg?t=%E8%AE%BE%E6%A4%AD%E5%9C%86%E7%9A%84%E6%96%B9%E7%A8%8B%E4%B8%BAX%E5%B9%B3%E6%96%B9%2BY%E5%B9%B3%E6%96%B9%2F4%3D1%2C%E8%BF%87M%EF%BC%880%2C1%EF%BC%89%E7%9A%84%E7%9B%B4%E7%BA%BF%E4%BA%A4%E6%A4%AD%E5%9C%86%E4%BA%8EAB%E4%B8%A4%E7%82%B9%2CO%E4%B8%BA%E5%9D%90%E6%A0%87%E5%8E%9F%E7%82%B9%2COP%E5%90%91%E9%87%8F%3D1%2F2%28OA%E5%90%91%E9%87%8F%2BOB%E5%90%91%E9%87%8F%29%2C%E5%BD%93L%E7%BB%95%E7%82%B9M%E6%97%8B%E8%BD%AC%E6%97%B6%2C%E6%B1%82%E5%8A%A8%E7%82%B9P%E7%9A%84%E8%BD%A8%E8%BF%B9%E6%96%B9%E7%A8%8B)
设椭圆的方程为X平方+Y平方/4=1,过M(0,1)的直线交椭圆于AB两点,O为坐标原点,OP向量=1/2(OA向量+OB向量),当L绕点M旋转时,求动点P的轨迹方程
设椭圆的方程为X平方+Y平方/4=1,过M(0,1)的直线交椭圆于AB两点,O为坐标原点,OP向量=1/2(OA向量+OB向量),当L绕点M旋转时,求动点P的轨迹方程
设椭圆的方程为X平方+Y平方/4=1,过M(0,1)的直线交椭圆于AB两点,O为坐标原点,OP向量=1/2(OA向量+OB向量),当L绕点M旋转时,求动点P的轨迹方程
E: x^2+y^2/4 = 1 (1)
M(0,1)
OP = (1/2)(OA+OB)
L: passing through M(0,1)
y = mx +c
1= c
ie
L: y = mx +1 (2)
Sub (2) into (1)
x^2 + (mx+1)^2/4 =1
4x^2 + (mx+1)^2 = 4
(4+m^2)x^2 + 2mx -3 =0
Let P be (x,y)
then
2x = -2m/(4+m^2) (3)
from (2)
y = mx+1
x = (y-1)/m (4)
Sub (4) into (1)
(y-1)^2/m^2 + y^2/4 = 1
4(y-1)^2 + m^2y^2 = 4m^2
(4+m^2)y^2 - 8y + 4(1-m^2) =0
then
2y = 8/(4+m^2)
4+m^2 = 4/y
m = √[4(1-y)/y] (5)
Sub (5) into (3)
2x = -2m/(4+m^2)
x = -√[4(1-y)/y]/ (4/y)
x^2 = [4(1-y)/y] / [4/y]^2
= y(1-y)/4
4x^2 = y(1-y)
P的轨迹方程:
4x^2 = y(1-y)