The equation of the plane passing through $(1,1,1)$ and $(1,-1,-1)$ and perpendicular to $2 x-y+z+5=0$ is :

Let the equation of plane passing through $(1,1,1)$ is $a(x-1)+b(y-1)+c(z-1)=0$ $\ldots$ (i) It is also passing through $(1,-1,-1)$. $\therefore \quad a(0)+b(-2)+c(-2)=0$ or $b+c=0$ $\ldots$ (ii) Since, Eq. (i) is perpendicular to the plane $2 x-y+z+5=0$ $\therefore \quad 2 a-b+c=0$ $\ldots$ (iii) Since, Eqs. (ii) and (iii) are identical $\therefore \quad \frac{a}{-2-2}=\frac{b}{-4-0}=\frac{c}{4} \Rightarrow \frac{a}{-4}=\frac{b}{-4}=\frac{c}{4}$ $\therefore$ Required equation of plane is $x+y-z-1=0$