已知离心率为1/2的椭圆C:x²/a²+y²/b²=1(a>b>0)经过点P(1.3/2),A1是椭圆C的右顶点.①求椭圆C的方程.②若直线l:x=my+2/7与椭圆C相交于A,B俩点,求证:向量A4A×向量A1B=0
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![已知离心率为1/2的椭圆C:x²/a²+y²/b²=1(a>b>0)经过点P(1.3/2),A1是椭圆C的右顶点.①求椭圆C的方程.②若直线l:x=my+2/7与椭圆C相交于A,B俩点,求证:向量A4A×向量A1B=0](/uploads/image/z/5181206-14-6.jpg?t=%E5%B7%B2%E7%9F%A5%E7%A6%BB%E5%BF%83%E7%8E%87%E4%B8%BA1%2F2%E7%9A%84%E6%A4%AD%E5%9C%86C%3Ax%26%23178%3B%2Fa%26%23178%3B%2By%26%23178%3B%2Fb%26%23178%3B%3D1%28a%3Eb%3E0%29%E7%BB%8F%E8%BF%87%E7%82%B9P%281.3%2F2%29%2CA1%E6%98%AF%E6%A4%AD%E5%9C%86C%E7%9A%84%E5%8F%B3%E9%A1%B6%E7%82%B9.%E2%91%A0%E6%B1%82%E6%A4%AD%E5%9C%86C%E7%9A%84%E6%96%B9%E7%A8%8B.%E2%91%A1%E8%8B%A5%E7%9B%B4%E7%BA%BFl%3Ax%3Dmy%2B2%2F7%E4%B8%8E%E6%A4%AD%E5%9C%86C%E7%9B%B8%E4%BA%A4%E4%BA%8EA%2CB%E4%BF%A9%E7%82%B9%2C%E6%B1%82%E8%AF%81%3A%E5%90%91%E9%87%8FA4A%C3%97%E5%90%91%E9%87%8FA1B%3D0)
已知离心率为1/2的椭圆C:x²/a²+y²/b²=1(a>b>0)经过点P(1.3/2),A1是椭圆C的右顶点.①求椭圆C的方程.②若直线l:x=my+2/7与椭圆C相交于A,B俩点,求证:向量A4A×向量A1B=0
已知离心率为1/2的椭圆C:x²/a²+y²/b²=1(a>b>0)经过点P(1.3/2),A1是椭圆C的右顶点.①求椭圆C的方程.②若直线l:x=my+2/7与椭圆C相交于A,B俩点,求证:向量A4A×向量A1B=0
已知离心率为1/2的椭圆C:x²/a²+y²/b²=1(a>b>0)经过点P(1.3/2),A1是椭圆C的右顶点.①求椭圆C的方程.②若直线l:x=my+2/7与椭圆C相交于A,B俩点,求证:向量A4A×向量A1B=0
①c/a=1/2,∴a=2c,b^2=3c^2,
∴椭圆x^2/(4c^2)+y^2/(3c^2)=1过点(1,3/2),
∴1/c^2=1,c^2=1,
∴椭圆的方程是x^2/4+y^2/3=1.①
②A1(2,0),把x=my+2/7②代入①*12,得3[m^2y^2+(4/7)my+4/49]+4y^2=12,
整理得(3m^2+4)y^2+(12/7)my-576/49=0,
设A(x1,y1),B(x2,y2),则y1+y2=-12m/[7(3m^2+4)],y1y2=-576/[49(3m^2+4)],
∴向量A1A*A1B=(x1-2,y1)*(x2-2,y2)=(x1-2)(x2-2)+y1y2
=(my1-12/7)(my2-12/7)+y1y2(由②)
=(m^2+1)y1y2-(12/7)m(y1+y2)+144/49,
=[-576(m^2+1)+144m^2+144(3m^2+4)]/[49(3m^2+4)]
=0.