The equation of the plane passing through the point $(1,2,2)$ and perpendicular to the planes $x-y+2z=3$ and $2x-2y+z+12=0$ is:

The distance between the planes $x + 2y + 3z + 7 = 0$ and $2x + 4y + 6z + 7 = 0$ is

In a tetrahedron $LMNO$,edges $ML, MN$ and $MO$ are mutually perpendicular. If the lengths of the altitudes drawn from $O, L$ and $N$ to their opposite faces are $1, 2$ and $3$ units respectively,then the length of the altitude drawn from $M$ to the face $LNO$ is:

$A$ variable plane at a constant distance $p$ from the origin meets the coordinate axes at points $A, B, C$. Through these points,planes are drawn parallel to the coordinate planes. Find the locus of their point of intersection.

$A$ variable plane is at a distance of $6$ units from the origin. If it meets the coordinate axes in $A, B$,and $C$,then the equation of the locus of the centroid of the $\triangle ABC$ is

$A$ plane $\pi_1$ passing through the point $3 \hat{i}-7 \hat{j}+5 \hat{k}$ is perpendicular to the vector $\hat{i}+2 \hat{j}-2 \hat{k}$ and another plane $\pi_2$ passing through the point $2 \hat{i}+7 \hat{j}-8 \hat{k}$ is perpendicular to the vector $3 \hat{i}+2 \hat{j}+6 \hat{k}$. If $p_1$ and $p_2$ are the perpendicular distances from the origin to the planes $\pi_1$ and $\pi_2$ respectively,then $p_1-p_2=$