If P and the origin are the point of intersection of the parabolas ${\mathrm{y}}^2=32\mathrm{x}$ and $2{\mathrm{x}}^2=27\mathrm{y},$ and if $\mathrm{\theta }$ is the acute angle between these curves at P then $5\sqrt{\mathrm{tan\theta }}=$
 

$$\begin{gathered} {\mathrm{y}}^2=32\mathrm{x} \\ 4\mathrm{a}=32 \\ \Rightarrow \overline{) \mathrm{a}=8 } \\ and 2{\mathrm{x}}^2=27\mathrm{y} \\ \Rightarrow {\mathrm{x}}^2=\frac{27}{2}\mathrm{y} \end{gathered}$$

                        $$\begin{gathered} \mathrm{compare} \mathrm{with} {\mathrm{x}}^2=4\mathrm{by} \\ \Rightarrow 4\mathrm{b}=27/2 \\ \Rightarrow \mathrm{b}=27/8 \\ If \theta is angle between curves then \\ \tan \mathrm{\theta }=\left(\frac{3{\mathrm{a}}^{1/3} {\mathrm{b}}^{1/3}}{2({\mathrm{a}}^{2/3}+{\mathrm{b}}^{2/3})}\right) \end{gathered}$$

$$\begin{gathered} \tan \mathrm{\theta }=\frac{3.\cancel{2 } {\displaystyle \frac{3}{\cancel{2}}}}{2\left(4+{\displaystyle \frac{9}{4}}\right)}=\frac{9}{{\displaystyle \frac{25}{2}}} \\ \tan \mathrm{\theta }=\frac{18}{25} \\ \sqrt{\tan \mathrm{\theta }}=\frac{3\sqrt{2}}{5} \\ 5\sqrt{\tan \mathrm{\theta }}=3\sqrt{2} \end{gathered}$$