$(b)$ Velocity $v$ of a particle moving around a curved path can be resolved into $\operatorname{radial}\left(v_{r}=\frac{d r}{ d t}\right)$ and tangential (or transverse, $\left.v_{\theta}=r \frac{d \theta}{d t}\right)$ velocity components.
Angle $\alpha$ between velocity $v$ and tangential velocity $v _{r}$ is given by
$\tan \alpha=\frac{v_{\theta}}{v_{r}}$
$\Rightarrow \quad \tan \alpha=\frac{\left(r \frac{d \theta}{d t}\right)}{\left(\frac{d r}{d t}\right)}=r \cdot\left(\frac{d \theta}{d r}\right)$
$=\frac{r}{(d r / d \theta)}$
Here, given $\frac{d r}{d \theta}=r$
$\therefore \tan \alpha=\frac{r}{r}=1$
$\Rightarrow \alpha=\tan ^{-1}(1)=45^{\circ}$