If the foot of the perpendicular from $(0,0,0)$ to the plane is $(1,2,2)$,then the equation of the plane is

$A$ plane passes through the point $A(2, 1, -3).$ If the distance of this plane from the origin is maximum,then its equation is

The equation of the plane passing through $(1,0,0)$ and $(0,1,0)$ and making an angle of $45^{\circ}$ with the plane $x+y-3=0$ is:

$A$ plane $\Pi$ passes through the points $A=(0,0,2)$,$B=(1,0,1)$,and $C=(3,1,1)$. If the plane $\Pi$ makes angles $\alpha$ and $\beta$ with the $XY$ and $XZ$-coordinate planes respectively,then $\sin^2 \alpha + \sin^2 \beta =$

If the vector equation of the plane $\bar{r}=(2 \hat{i}+\hat{k})+\lambda \hat{i}+\mu(\hat{i}+2 \hat{j}-3 \hat{k})$ in scalar product form is given by $\bar{r} \cdot(3 \hat{j}+2 \hat{k})=\alpha$,then $\alpha=$

The equation of the plane passing through the point $(2, 3, 4)$ and parallel to the plane $x + 2y + 4z = 5$ is: