AMC 12 2004b Problem 10

An annulus is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c$, with $b>c$. Let $OX$ be a radius of the larger circle, let $XZ$ be tangent to the smaller circle at $Z$, and let $OY$ be the radius of the larger circle that contains $Z$. Let $a=XZ$, $d=YZ$, and $e=XY$. What is the area of the annulus?

import graph;
unitsize(1.5cm);
defaultpen(0.8);
real r1=1.5, r2=2.5;
pair O=(0,0);
path inner=Circle(O,r1), outer=Circle(O,r2);
pair Y=(0,r2), Z=(0,r1), X=intersectionpoint( Z--(Z+(10,0)), outer );
filldraw(outer,lightgray,black);
filldraw(inner,white,black);
draw(X--O--Y); draw(Y--X--Z);
label("$O$",O,SW);
label("$X$",X,E);
label("$Y$",Y,N);
label("$Z$",Z,SW);
label("$a$",X--Z,N);
label("$b$",0.25*X,SE);
label("$c$",O--Z,E);
label("$d$",Y--Z,W);
label("$e$",Y*0.65 + X*0.35,SW);
defaultpen(0.5);
dot(O); dot(X); dot(Z); dot(Y);

$\mathrm{(A) \ } \pi a^2 \mathrm{(B) \ } \pi b^2 \mathrm{(C) \ } \pi c^2 \mathrm{(D) \ } \pi d^2 \mathrm{(E) \ } \pi e^2$