數學問題:幾何方程

Let (x,y) be any point on the required boundary.

1. A(4,0), B(8cost, 8sint), AB vector //(2cost -1, 2sint), mid of AB=(2+4cost,4sint)

(x,y) satisfies (2cost -1)(x-2-4cost)+2sint (y-4sint)=0

or cost x + sint y= (6+x)/2 (for some t) -----(*)

2. lim(u->t) {[(2cosu-1)x+2sinu y]-[(2cost -1)x+2sint y] }/(u-t) =0, then

sint x - cost y=0 -----(**)

Note: the intersection of cost x+ sint y=(6+x)/2 and cosu x+sinu y=(6+x)/2

is (x,y) when u->t

From (*)and(**), then x²+y²=(6+x)² /4 thus (x-2)²/16+ y²/12=1

Ans: (x-2)²/16+y²/12=1