Question

Let $\alpha x+\beta y+y z=1$ be the equation of a plane passing through the point $(3,-2,5)$ and perpendicular to the line joining the points $(1,2,3)$ and $(-2,3,5)$. Then the value of $\alpha \beta y$ is equal to ____

Answer 1

Given Equation is not equation of plane as $y y$ is present. If we consider $y$ is $\gamma$ then answer would be 6 .
Normal vector of plane $=3 \hat{i}-\hat{j}-2 \hat{k}$
Plane $: 3 x - y -2 z +\lambda=0$
Point $(3,-2,5)$ satisfies the plane
$ \lambda=-1$
$ 3 x-y-2 z=1$
$ \alpha \beta y=6$