Circles problem again help out plzzzz

Let A and B be points on the x-axis which are intercepts of circles x^2+y^2=1,another circles with centre at B and variable radius intersects the first circle at C above the x-axis and the line segment AB and D.Find the maximum area of triangle BCD?

Let the equation of circle B be
(x-1)^2+y^2=r^2------------(1)
where r is the radius.
x^2+y^2=1------------------(2)
Solving (1) & (2) simultaneously, get
x=1-(r^2)/2
y=rsqrt(4-r^2)/2 (take +ve y only)
The area of tri.BCD, A=
0.5BDy=
0.5ry=
0.5r[rsqrt(4-r^2)/2]=>
A=(r^2/4)sqrt(4-r^2)=>
dA/dr=(8r-3r^3)/[4sqrt(4-r^2)]
dA/dr=0=>
r(8-3r^2)=0=>
r=0(rejected) or r=2sqrt(2)/3
d^2A/dr^2[2sqrt(2)/3]<0=>
A(max.)=8sqrt(4-8/3)/12=>
.............=4sqrt(3)/9

Are A and B BOTH on x-axis? How can it be. Please check.