设动点M到A(―5, 0)的距离与它到B(5, 0)

设M(x,y)
则MA=√[(x+5)^2+y^2]
MB=√[(x-5)^2+y^2]
所以√[(x+5)^2+y^2]=6+√[(x+5)^2+y^2]
两边平方
x^2+10x+25+y^2=36+12√[(x+5)^2+y^2]+x^2-10x+25+y^2
5x-9=3√[(x+5)^2+y^2]
两边平方
(5x-9)^2=9(x+5)^2+9y^2
(5x-9)^2-(3x+15)^2=y^2
(5x-9+3x+15)(5x-9-3x-15)=y^2
(8x+6)(2x-24)-y^2=0
16x^2-180x-144-y^2=0

y^2=(144-9x^2)/16>=0
所以144-9x^2>=0
x^2<=16
-4<=x<=4
x+y^2=x+(144-9x^2)/16
=-9x^2/16+x+9
=(-9/16)(x-8/9)^2+85/9
-4<=x<=4
所以x=8/9,x+y^2最大=85/9
x=-4，x+y^2最小=-4

1.x^2/9-y^2/16=0
2.85/9,-4
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