Question

与园（x+1)²+y²=1外切,且与圆（x-1)²+y²=9内切的动圆圆心的轨迹

Answer 1

（x+1)²+y²=1的圆心坐标(-1,0),半径1

（x-1)²+y²=9的圆心坐标(1,0),半径3

令与园（x+1)²+y²=1外切,且与圆（x-1)²+y²=9内切的圆的圆心坐标为(x,y),半径r

∵与园（x+1)²+y²=1外切,∴√{ (x-(-1))^2+(y-0)^2 } = r+1

∵与园（x-1)²+y²=3内切,∴√{ (x-1)^2+(y-0)^2 } = 3-r

两式相加得：√{ (x-(-1))^2+(y-0)^2 } + √{ (x-1)^2+(y-0)^2 } = 4

即：√{ (x+1)^2+y^2 } + √{ (x-1)^2+y^2 } = 4

√{ (x+1)^2+y^2 } =4- √{ (x-1)^2+y^2 }

(x+1)^2+y^2 = 16 - 8√{ (x-1)^2+y^2 } + (x-1)^2+y^2

2√{ (x-1)^2+y^2 } = 4 -x

4{ (x-1)^2+y^2 } = 16-8x+x^2

4x^2-8x+4+y^2 = 16-8x+x^2

3x^2+y^2 = 12

x^2/4 + y^2/12 = 1