Find the intercepts made by the plane $\bar{r} \cdot (2, -3, 4) = 12$ on the coordinate axes.

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 $\frac{d}{b+c}$.

If $-2, \frac{4}{3}, \frac{-4}{5}$ are the intercepts made by a plane on $X, Y, Z$-axes respectively,then the direction cosines of a normal to this plane are

$A$ plane passing through a point $(2,2,2)$ cuts the positive semi-axes at $A$,$B$,and $C$. If $P(\alpha, \beta, \gamma)$ is the centroid of the tetrahedron $OABC$ (where $O$ is the origin),then select the correct option.

In the following cases,determine whether the given planes are parallel or perpendicular,and in case they are neither,find the angle between them:
$4x + 8y + z - 8 = 0$ and $y + z - 4 = 0$

$A$ point on the plane that passes through the points $(1, -1, 6)$, $(0, 0, 7)$ and is perpendicular to the plane $x - 2y + z = 6$ is