椭圆的数学题,已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),经过点M(1,3/2),其离心率为1/2设直线l:y=kx+m(|k|
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![椭圆的数学题,已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),经过点M(1,3/2),其离心率为1/2设直线l:y=kx+m(|k|](/uploads/image/z/9969014-38-4.jpg?t=%E6%A4%AD%E5%9C%86%E7%9A%84%E6%95%B0%E5%AD%A6%E9%A2%98%2C%E5%B7%B2%E7%9F%A5%E6%A4%AD%E5%9C%86x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2%3D1%28a%3Eb%3E0%29%2C%E7%BB%8F%E8%BF%87%E7%82%B9M%EF%BC%881%2C3%2F2%29%2C%E5%85%B6%E7%A6%BB%E5%BF%83%E7%8E%87%E4%B8%BA1%2F2%E8%AE%BE%E7%9B%B4%E7%BA%BFl%EF%BC%9Ay%3Dkx%2Bm%28%7Ck%7C)
椭圆的数学题,已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),经过点M(1,3/2),其离心率为1/2设直线l:y=kx+m(|k|
椭圆的数学题,
已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),经过点M(1,3/2),其离心率为1/2设直线l:y=kx+m(|k|
椭圆的数学题,已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),经过点M(1,3/2),其离心率为1/2设直线l:y=kx+m(|k|
离心率e = c/a = 1/2 得b = √(3)/2 * a
(1,3/2)在椭圆上,即1 / a^2 + 9/(4*3/4 * a^2) = 1,解得a = 2
所以椭圆方程为x^2 / 4 + y^2 / 3 = 1.
设A(x1,y1),B(x2,y2)
因为向量OP = OA + OB
所以P点坐标为(x3,y3) = (x1+x2,y1 + y2)
k = (y1-y2)/(x1-x2)
将x1^2 / 4 + y1^2 / 3 = 1
与x2^2 / 4 + y2^2 / 3 = 1
相减(用平方差公式),得
(x1 – x2)(x1 + x2)/4 + (y1 – y2)(y1+y2)/3 = 0
所以 (y1 + y2) / (x1 + x2) = -3(x1-x2)/4(y1-y2) = -3/4k
由-1/2 < k < 1/2可得 y3 / x3 > 3/2 或 y3 / x3 < -3/2
设y3 = d * x3,d < -3/2 或d > 3/2
因为P在椭圆上,所以(1/4 + d^2/3) * x3^2 = 1
|OP|^2 = (1+d^2) * x3^2 = (1+d^2)/(1/4 + d^2/3) = 3 + 3/(3+4d^2)
由前面d^2 > 9/4知,3≤|OP|^2 <3 + 3/(3 + 4*9/4) = 13/4
所以|OP|的取值范围是[√(3),√(13)/2)