Question

Let $P_{1}: \vec{r} \cdot(2 \hat{i}+\hat{j}-3 \hat{k})=4$ be a plane. Let $P_{2}$ be another plane which passes through the points (2,$3,2)(2,-2,-3)$ and $(1,-4,2)$. If the direction ratios of the line of intersection of $P_{1}$ and $P_{2}$ be 16, $\alpha, \beta$, then the value of $\alpha+\beta$ is equal to________.

Answer 1

$P_{1}: \vec{r} \cdot(2 \hat{i}+\hat{j}-3 \hat{k})=4$
$P_{1}: 2 x+y-3 z=4$
$P_{2} \begin{vmatrix}x-2 & y+3 & z-2 \\0 & 1 & -5 \\-1 & -1 & 0\end{vmatrix}=0 $
$\Rightarrow-5 x+5 y+z+23=0$
Let a, b, c be the d'rs of line of intersection
Then $ a =\frac{16 \lambda}{15} ; b =\frac{13 \lambda}{15} ; c =\frac{15 \lambda}{15} $
$\therefore \alpha=13: \beta=15$