已知椭圆x^2/a^2+y^2/b^2=1(a>b>0)的离心率e=√6/3,过点A(0,b)和B(a,0)的直线与原点的距离为√3/2拜托各已知定点E(-1,0),若直线y=kx+2(k≠0)与椭圆交于C、D两点,问:是否存在k的值,使以CD为直径的圆过E点?
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![已知椭圆x^2/a^2+y^2/b^2=1(a>b>0)的离心率e=√6/3,过点A(0,b)和B(a,0)的直线与原点的距离为√3/2拜托各已知定点E(-1,0),若直线y=kx+2(k≠0)与椭圆交于C、D两点,问:是否存在k的值,使以CD为直径的圆过E点?](/uploads/image/z/5370560-8-0.jpg?t=%E5%B7%B2%E7%9F%A5%E6%A4%AD%E5%9C%86x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2%3D1%28a%3Eb%3E0%29%E7%9A%84%E7%A6%BB%E5%BF%83%E7%8E%87e%3D%E2%88%9A6%2F3%2C%E8%BF%87%E7%82%B9A%280%2Cb%29%E5%92%8CB%28a%2C0%29%E7%9A%84%E7%9B%B4%E7%BA%BF%E4%B8%8E%E5%8E%9F%E7%82%B9%E7%9A%84%E8%B7%9D%E7%A6%BB%E4%B8%BA%E2%88%9A3%2F2%E6%8B%9C%E6%89%98%E5%90%84%E5%B7%B2%E7%9F%A5%E5%AE%9A%E7%82%B9E%28-1%2C0%29%2C%E8%8B%A5%E7%9B%B4%E7%BA%BFy%3Dkx%2B2%28k%E2%89%A00%29%E4%B8%8E%E6%A4%AD%E5%9C%86%E4%BA%A4%E4%BA%8EC%E3%80%81D%E4%B8%A4%E7%82%B9%2C%E9%97%AE%EF%BC%9A%E6%98%AF%E5%90%A6%E5%AD%98%E5%9C%A8k%E7%9A%84%E5%80%BC%2C%E4%BD%BF%E4%BB%A5CD%E4%B8%BA%E7%9B%B4%E5%BE%84%E7%9A%84%E5%9C%86%E8%BF%87E%E7%82%B9%3F)
已知椭圆x^2/a^2+y^2/b^2=1(a>b>0)的离心率e=√6/3,过点A(0,b)和B(a,0)的直线与原点的距离为√3/2拜托各已知定点E(-1,0),若直线y=kx+2(k≠0)与椭圆交于C、D两点,问:是否存在k的值,使以CD为直径的圆过E点?
已知椭圆x^2/a^2+y^2/b^2=1(a>b>0)的离心率e=√6/3,过点A(0,b)和B(a,0)的直线与原点的距离为√3/2拜托各
已知定点E(-1,0),若直线y=kx+2(k≠0)与椭圆交于C、D两点,问:是否存在k的值,使以CD为直径的圆过E点?请说明理由.
已知椭圆x^2/a^2+y^2/b^2=1(a>b>0)的离心率e=√6/3,过点A(0,b)和B(a,0)的直线与原点的距离为√3/2拜托各已知定点E(-1,0),若直线y=kx+2(k≠0)与椭圆交于C、D两点,问:是否存在k的值,使以CD为直径的圆过E点?
依题意,得:椭圆方程为x^2/3+y^2=1 设CD的坐标分别是(x1,y1),(x2,y2) EC=(x1+1,y1),ED=(x2+1,y2),EC,ED是向量 若E在以CD为直径的圆的圆周上,则有EC*ED=0 (x1+1)(x2+1)+y1y2=0 x1x2+(x1+x2)+1+y1y2=0 x1x2+(x1+x2)+1+(kx1+2)(kx2+2) (k+1)x1x2+(2k+1)(x1+x2)+5=0 将y=kx+2代入椭圆方程 x/3+(kx+2)=1 (1/3+k)x+4kx+3=0 x1+x2=-4k/(1/3+k),x1x2=3/(1/3+k) 代入化简得 3(k+1)-4k(2k+1)+5(1/3+k)=0 (14/3)-4k=0 k=7/6