If $(1,5,35),(7,5,5),(1, \lambda, 7)$ and $(2 \lambda, 1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is:
Correct Option: , 4
Let $\mathrm{P}(1,5,35), \mathrm{Q}(7,5,5), \mathrm{R}(1, \lambda, 7), \mathrm{S}(2 \lambda, 1,2)$
$\begin{array}{ll}\overrightarrow{\mathrm{PQPR}} & \overrightarrow{\mathrm{PS}}]=0\end{array}$
$\left|\begin{array}{ccc}6 & 0 & -30 \\ 0 & \lambda-5 & -28 \\ 2 \lambda-1 & -4 & -33\end{array}\right|=0$
$\left|\begin{array}{ccc}1 & 0 & -5 \\ 0 & \lambda-5 & -28 \\ 2 \lambda-1 & -4 & -33\end{array}\right|=0$
$\{-33 \lambda+165-112\}+5(\lambda-5)(2 \lambda-1)=0$
$53-33 \lambda+5\left\{2 \lambda^{2}-11 \lambda+5\right\}=0$
$16 \lambda^{2}-88 \lambda+78=0$
$5 \lambda^{2}-44 \lambda+39=0<_{\lambda_{2}}^{\lambda_{1}}$
$\Rightarrow \lambda_{1}+\lambda_{2}=44 / 5$
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