$\text{ The angle between the parabolas }{y}^2=4ax\text{ and }{x}^2=4ay\text{ at their point of intersection other than the origin.}$

$\begin{array}{l}\text{ Pt of intersection of }{y}^2=4ax\mathrm{\&}{x}^2=4ay\\ \text{ are }(0,0)(4\mathrm{a},4\mathrm{a})\\ 2y\frac{dy}{dx}=4a\Rightarrow \frac{dy}{dx}=\frac{2a}{y}\\ m_1={\left(\frac{dy}{dx}\right)}_{(4a,4a)}=\frac{2a}{4a}=\frac{1}{2}\\ m_2{=}_{(4a,4a)}=\frac{2x}{4a}=\frac{8a}{4a}=2\end{array}$

$\left|\tan \theta =\frac{2-\frac{1}{2}}{1+1}\right|=\frac{3}{4}$

$\tan \theta =\frac{3}{4}\Rightarrow \theta ={\tan }^{-1}\left(\frac{3}{4}\right)$