Question:
Suppose the line $\frac{x-2}{\alpha}=\frac{y-2}{-5}=\frac{z+2}{2}$ lies on the plane $x+3 y-2 z+\beta=0$. Then $(\alpha+\beta)$ is equal to_____.
Solution:
Point $(2,2,-2)$ also lies on given plane
So $2+3 \times 2-2(-2)+\beta=0$
$\Rightarrow 2+6+4+\beta=0 \Rightarrow \beta=-12$
Also $\alpha \times 1-5 \times 3+2 \times-2=0$
$\Rightarrow \alpha-15-4=0 \Rightarrow \alpha=19$
$\therefore \alpha+\beta=19-12=7$