1.
设M、N分别为(x1,mx1)、(x2,-mx2),P(x,y)
∵向量OP的二倍=向量OM+向量ON
∴P是MN的中点
∴x1+x2=2x
mx1-mx2=2y,x1-x2=2y/m
|MN|=√[(x1-x2)²+(mx1+mx2)²]=2
∴(x1-x2)²+(mx1+mx2)²=4
即(x1-x2)²+m²(x1+x2)²=4
∵x1+x2=2x,x1-x2=2y/m
∴(2x)²+(2y/m)²=4
即x² + y²/m² = 1
∴当0<m<1时,曲线C是焦点在x轴上的椭圆.
当m=1时,曲线C是以原点为圆心,1为半径的圆.
当m>1时,曲线C是焦点在y轴上的椭圆.
2.
当m>1时,曲线C是焦点在y轴上的椭圆.设A(x1,y1)、B(x2,y2)
y=kx+1
x² + y²/m² = 1
联立得(m²+k²)x²+2kx+1-m²=0
∠AOB是锐角
cos∠AOB>0
cos∠AOB=(向量OA·向量OB)/(|OA||OB|)>0
∴向量OA·向量OB>0
即x1x2+y1y2>0
x1x2=(1-m²)/(m²+k²)
y1y2
=(kx1+1)(kx2+1)
=k²x1x2 + k(x1+x2) + 1
=k²(1-m²)/(m²+k²) - 2k²/(m²+k²) +1
x1x2+y1y2
=(1-m²)/(m²+k²) + k²(1-m²)/(m²+k²) - 2k²/(m²+k²) +1
=(1-m²+k²-k²m²-2k²+m²+k²)/(m²+k²)
=(1-k²m²)/(m²+k²) >0
即1-k²m²>0
k²<1/m²
∴-1/m<k<1/m