$\overline{a}=\lambda \widehat{i}+\mu \widehat{j}+4\widehat{k}$ , $\overline{b}=-2\widehat{i}+4\widehat{j}-2\widehat{k}$  and $\overrightarrow{c}=2\widehat{i}+3\widehat{j}+\widehat{k}$  are coplanar and the projection of $\overline{a}$  on the vector $\overline{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda +\mu$  is equal to 

Given equations are coplanar
So, $\left|\begin{array}{ccc}\lambda & \mu & 4\\ -2& 4& -2\\ 2& 3& 1\end{array}\right|=0$  
 $\lambda \left(4+6\right)-\mu \left(-2+4\right)+4\left(-6-8\right)=0$ 

$10\lambda -2\mu -56-0\Rightarrow 5\lambda -\mu -28=0\to \left(1\right)$
Given projection of  $\overline{a}$ on $\overline{b}$  is $\sqrt{54}$  
$\Rightarrow \frac{\overline{a}.\overline{b}}{\left|\overline{b}\right|}=\sqrt{54}$ 
$\Rightarrow \frac{-2\lambda +4\mu -8}{\sqrt{24}}=\sqrt{54}$ 
 $\Rightarrow -\lambda +2\mu =22\to \left(2\right)$
Solving (1) & (2)
$\lambda =\frac{26}{3}, \mu =\frac{46}{3}$
 $\lambda +\mu =24$