1.已知A(-1/2,0),B是圆F:(x-1/2)^2+y^2=4上一动点,线段AB的垂直平分线交BF于点P,求动点P的轨迹方程2.直线l:y=kx+1与椭圆C:ax^2+y^2=2(a>1)交于A,B两点,以OA,OB为邻边作平行四边形OAPB(1)若k=1,且四
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![1.已知A(-1/2,0),B是圆F:(x-1/2)^2+y^2=4上一动点,线段AB的垂直平分线交BF于点P,求动点P的轨迹方程2.直线l:y=kx+1与椭圆C:ax^2+y^2=2(a>1)交于A,B两点,以OA,OB为邻边作平行四边形OAPB(1)若k=1,且四](/uploads/image/z/4972944-48-4.jpg?t=1.%E5%B7%B2%E7%9F%A5A%EF%BC%88-1%2F2%2C0%EF%BC%89%2CB%E6%98%AF%E5%9C%86F%EF%BC%9A%EF%BC%88x-1%2F2%EF%BC%89%5E2%2By%5E2%3D4%E4%B8%8A%E4%B8%80%E5%8A%A8%E7%82%B9%2C%E7%BA%BF%E6%AE%B5AB%E7%9A%84%E5%9E%82%E7%9B%B4%E5%B9%B3%E5%88%86%E7%BA%BF%E4%BA%A4BF%E4%BA%8E%E7%82%B9P%2C%E6%B1%82%E5%8A%A8%E7%82%B9P%E7%9A%84%E8%BD%A8%E8%BF%B9%E6%96%B9%E7%A8%8B2.%E7%9B%B4%E7%BA%BFl%EF%BC%9Ay%3Dkx%2B1%E4%B8%8E%E6%A4%AD%E5%9C%86C%3Aax%5E2%2By%5E2%3D2%28a%3E1%29%E4%BA%A4%E4%BA%8EA%2CB%E4%B8%A4%E7%82%B9%2C%E4%BB%A5OA%2COB%E4%B8%BA%E9%82%BB%E8%BE%B9%E4%BD%9C%E5%B9%B3%E8%A1%8C%E5%9B%9B%E8%BE%B9%E5%BD%A2OAPB%EF%BC%881%EF%BC%89%E8%8B%A5k%3D1%2C%E4%B8%94%E5%9B%9B)
1.已知A(-1/2,0),B是圆F:(x-1/2)^2+y^2=4上一动点,线段AB的垂直平分线交BF于点P,求动点P的轨迹方程2.直线l:y=kx+1与椭圆C:ax^2+y^2=2(a>1)交于A,B两点,以OA,OB为邻边作平行四边形OAPB(1)若k=1,且四
1.已知A(-1/2,0),B是圆F:(x-1/2)^2+y^2=4上一动点,线段AB的垂直平分线交BF于点P,求动点P的
轨迹方程
2.直线l:y=kx+1与椭圆C:ax^2+y^2=2(a>1)交于A,B两点,以OA,OB为邻边作平行四边形OAPB(1)若k=1,且四边形为矩形,求a的值(2)若a=2,当k变化时,求点P的轨迹方程
1.已知A(-1/2,0),B是圆F:(x-1/2)^2+y^2=4上一动点,线段AB的垂直平分线交BF于点P,求动点P的轨迹方程2.直线l:y=kx+1与椭圆C:ax^2+y^2=2(a>1)交于A,B两点,以OA,OB为邻边作平行四边形OAPB(1)若k=1,且四
(1) 垂直平分线上的点到A,B的距离相等,
∴PA=PB
半径=2=BF=PB+PF=PA+PF
可见P点到(-1/2,0)和(1/2,0)的距离和为定值2,
P轨迹是椭圆
c=1/2,2a=2,
则b^2=3/4
P的轨迹方程是:x^2+4y^2/3=1
(2)
k=1
y=x+1代入ax^2+y^2=2得
(a+1)x^2+2x-1=0
设A(x1,y1),B(x2,y2)
四边形为矩形
∴OA⊥OB
∴x1x2+y1y2=0
x1x2+x1x2+x1+x2+1=0
2x1x2+x1+x2+1=0
-2/(a+1)-2/(a+1)+1=0
∴a=3
第二问
2x^2+y^2=2
y=kx+1
2x^2+(kx+1)^2=2
(2+k^2)x^2+2kx-1=0
x1+x2=-2k/(2+k^2)
y1+y2=-2k^2/(2+k^2)+2=4/(2+k^2)
Px=(x1+x2)=-2k/(2+k^2)
Py=(y1+y2)=4/(2+k^2)
Px=(4Px/Py)/(2+(2Px/Py)^2)
P(x,y)的轨迹方程
x=(4x/y)/(2+(2x/y)^2)
2+4x^2/y^2=4/y
y^2+2x^2=2y
即2x^2+(y-1)^2=1