Question

高二数学双曲线

设点P到M（-1,0）、N（1,0）距离之差为2m，到x轴和y轴距离之比为2，求m的取值范围

Answer 1

|y|/|x|=2
|y|=2|x|
则y²=4x²

双曲线c=1
2a=2m则a=m
b²=1-m²
x²/m²-y²/(1-m²)=1
则x²/m²-4x²/(1-m²)=1
(1-m²-4m²)x²=m²(1-m²)
顶点(±m,0)
所以x²>=m²
则x²=m²(1-m²)/(1-m²-4m²)>=m²
(1-m²)/(1-5m²)-1>=0
4m²(1-5m²)>=0
0<=m²<=1/5
所以0<m<=√5/5

Answer 2

y=-(x^2-120x)-1200
=-[x^2-120x+(120/2)^2-(120/2)^2]-1200
=-(x^2-120x+3600-3600)-1200
=-(x^2-120x+3600)+3600-1200
=-(x-60)^2+2400