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

Answer: 28

$P_1:\vec{r}\cdot (2\hat{i}+\hat{j}-3\hat{k})=4$
$P_1:2x+y-3z=4$
$P_2\left|\begin{array}{ccc}x-2& y+3& z-2\\ 0& 1& -5\\ -1& -1& 0\end{array}\right|=0$
$\Rightarrow -5x+5y+z+23=0$
Let a, b, c be the d'rs of line of intersection
Then $a=\frac{16\lambda }{15};b=\frac{13\lambda }{15};c=\frac{15\lambda }{15}$
$\therefore \alpha =13:\beta =15$