Take points $x, y\in b$ which minimize (correspondingly maximize) the function $\mathop{\rm dist}_a$. Let $\bar x$ and $\bar y\in a$ be the closest points to $x$ and $y$ correspondingly.

Clearly $\bar x=a(t)$, $x=b(t)$ and $\bar y=a(\tau)$, $y=b(\tau)$ for some $t$ and $\tau$ and $\theta(t)=\theta(\tau)=\tfrac\pi2$.

If $a$ and $b$ are not concentric circles then you get only these two values.

Take points $x, y\in b$ which minimize (correspondingly maximize) the function $\mathop{\rm dist}_a$. Let $\bar x$ and $\bar y\in a$ be the closest points to $x$ and $y$ correspondingly.

Clearly $\bar x=a(t)$, $x=b(t)$ and $\bar y=a(\tau)$, $y=b(\tau)$ for some $t$ and $\tau$ and $\theta(t)=\theta(\tau)=\tfrac\pi2$.

If $a$ and $b$ are non-concentric circles then you get only these two values.

Take points $x, y\in b$ which minimize (correspondingly maximize) the function $\mathop{\rm dist}_a$. Let $\bar x$ and $\bar y\in a$ be the closest points to $x$ and $y$ correspondingly.

Clearly $\bar x=a(t)$, $x=b(t)$ and $\bar y=a(\tau)$, $y=b(\tau)$ for some $t$ and $\tau$ and $\theta(t)=\theta(\tau)=\tfrac\pi2$.

Take points $x, y\in b$ which minimize (correspondingly maximize) the function $\mathop{\rm dist}_a$. Let $\bar x$ and $\bar y\in a$ be the closest points to $x$ and $y$ correspondingly.

Clearly $\bar x=a(t)$, $x=b(t)$ and $\bar y=a(\tau)$, $y=b(\tau)$ for some $t$ and $\tau$ and $\theta(t)=\theta(\tau)=\tfrac\pi2$.

If $a$ and $b$ are not concentric circles then you get only these two values.