SE-SKOLORNAS 1997 Problem 1

Let $AC$ be a diameter of a circle and $AB$ be tangent to the circle. The segment $BC$ intersects the circle again at $D$ . Show that if $AC = 1$ , $AB = a$ , and $CD = b$ , then $$\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}$$