If the plane $\frac{x}{3}+\frac{y}{2}-\frac{z}{4}=1$ cuts the coordinate axes at points $A, B$,and $C$,then the area of the triangle $ABC$ is

$A$ plane passes through a fixed point $A(1, -2, 3)$. If the locus of the foot of the perpendicular to it from the point $B(0, 1, 2)$ is $x^2 + y^2 + z^2 + \alpha x + \beta y + \gamma z + \delta = 0$,then the value of $\alpha + \beta + \gamma + \delta$ is

The direction cosines of the normal drawn to the plane passing through the points $(2,-1,5)$,$(1,-3,4)$,and $(5,2,1)$ are

The Cartesian equation of a plane parallel to the plane $\vec{r} \cdot(2 \hat{i}+3 \hat{j}-4 \hat{k})=1$ and at a distance of $2$ units from it is

If the foot of the perpendicular drawn from the point $(0, 0, 0)$ to the plane is $(4, -2, -5)$,then the equation of the plane 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 $p_1-p_2=$