Let PM be the perpendicular from the point $P (1, 2, 3)$ to XY-plane. If OP makes an angle $\theta$ with the positive direction of the Z-axis and OM makes an angle $\varphi$ with the positive direction of X-axis, where O is the origin, then $3\sqrt{5}\tan \theta +\tan \varphi$ is ______.  

$$\begin{gathered} \Rightarrow 1=r\sin \theta .\cos \varphi ,2=r\sin \theta \sin \varphi ,3=r\cos \theta \mathrm{...}\left(i\right) \\ \Rightarrow 1^2+2^2+3^2=r^2{\sin }^2\theta {\cos }^2\varphi +r^2{\sin }^2\theta {\sin }^2\varphi +r^2{\cos }^2\theta \\ \begin{array}{l}=r^2{\sin }^2\theta ({\cos }^2\varphi +{\sin }^2\varphi )+r^2{\cos }^2\theta \\ =r^2{\sin }^2\theta +r^2{\cos }^2\theta =r^2\end{array} \\ \Rightarrow r=\pm \sqrt{14} \end{gathered}$$

$\therefore$ From Eq. (i), we have 

$\begin{array}{l}\sin \theta \cos \varphi =+\frac{1}{\sqrt{14}},\\ \sin \theta \sin \varphi =\frac{2}{\sqrt{14}},\cos \theta =\frac{3}{\sqrt{14}}\end{array}$

(neglecting –ve sign as acute angles)

$$\begin{gathered} \therefore \frac{\sin \theta \sin \varphi }{\sin \theta \cos \varphi }=\frac{2}{1} \\ and \tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{\sqrt{5}}{3} \\ \Rightarrow \tan \varphi =2\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\tan \theta =\frac{\sqrt{5}}{3} \end{gathered}$$