Question

Let the lines
$L_1:\overrightarrow{r}=\lambda (\hat{i}+2\hat{j}+3\hat{k}),\lambda \in R$
$L_2:\vec{r}=(\hat{i}+3\hat{j}+\hat{k})+\mu (\hat{i}+\hat{j}+5\hat{k});\mu \in R$
intersect at the point $S$. If a plane $ax+by-z$ $+d=0$ passes through $S$ and is parallel to both the lines $L_1$ and $L_2$, then the value of $a+b+$ $d$ is equal to _________

Answer 1

Answer: 5

Both the lines lie in the same plane

$\therefore$ equation of the plane
$\left|\begin{array}{ccc}x& y& z\\ 1& 2& 3\\ 1& 1& 5\end{array}\right|=0$
$\Rightarrow 7x-2y-z=0$
$\therefore a+b+d=5$