Question

If the point $P(1,-2,1)$ and $Q({\lambda }^2,\lambda ,2)$ lies on opposite sides of the plane $x-2y+z=5,$ then the number of integral values of $\lambda$ is equal to

Answer 1

If points $P(1,-2,1)$ and $Q({\lambda }^2,\lambda ,2)$ lies on opposite sides of plane $x-2y+z-5=0$
$\Rightarrow (a{x}_1+b{x}_1+c{z}_1-d)\cdot (a{x}_2+b{x}_2+c{z}_2-d)<0$
$\Rightarrow (1+4+1-5)\cdot ({\lambda }^2-2\lambda +2-5)<0\Rightarrow {\lambda }^2-2\lambda -3<0\Rightarrow (\lambda +1)(\lambda -3)<0\Rightarrow \lambda \in (-1,3)$
So, integral values of $\lambda$ are :$0,1,2$