If $A(2,1,-1)$, $B(6,-3,2)$, and $C(-3,12,4)$ are the vertices of a triangle $ABC$, and the equation of the plane containing the triangle $ABC$ is $53x + by + cz + d = 0$, then find the value of $\frac{d}{b+c}$.

The foot of the perpendicular from the origin $O$ to a plane $P$,which meets the coordinate axes at points $A, B, C$,is $(2, a, 4)$,where $a \in N$. If the volume of the tetrahedron $OABC$ is $144 \text{ unit}^3$,then which of the following points does $NOT$ lie on the plane $P$?

The perpendicular distance of the point $(1, -1, 2)$ from the plane $x + 2y + z = 4$ is

The mirror image of the point $A(1, 2, 3)$ in a plane is $B\left(-\frac{7}{3}, -\frac{4}{3}, -\frac{1}{3}\right)$. Which of the following points lies on this plane?

The direction ratios of the normal to the plane passing through the origin and the line of intersection of the planes $x+2y+3z=4$ and $4x+3y+2z=1$ are . . . . . .

Find the vector and Cartesian equations of the plane that passes through the point $(1, 4, 6)$ and the normal vector to the plane is $\hat{i} - 2\hat{j} + \hat{k}$.