Find the minimum value of △ABE area, that is, the maximum value of △BEO area.

It can be seen that when BD is tangent to circle C and above the X axis, the area of △BEO is the largest.

Because BC=3, CD= 1.

So the root number 2 of DB=2

Let D(x, y)

With BD squared =(y-0)? +(x-2)？ =8

CD？ =(y-0)？ +(x+ 1)？ = 1

Simultaneous two forms, x=-2/3, y=2 root 2/3.

Then the straight BD equation is y=- root number 2x/4+ root number 2/2.

So the coordinates of point E are (0, root number 2/2).

So △ Abe area =△ Abe area -△BEO area.

=2*2/2-2* Root number 2/2/2

The root number of =2 is 2/2