Let the plane $ax+by+cz+d=0$ bisect the line joining the points $(4,-3,1)$ and $(2,3,-5)$ at right angles. If $a, b, c, d$ are integers,then the minimum value of $(a^2+b^2+c^2+d^2)$ is

If $\bar{r}=(2-\lambda+\mu) \hat{i}+(1-\mu) \hat{j}+(2-3 \lambda+2 \mu) \hat{k}$ is the vector equation of a plane,then the equivalent cartesian equation of the plane is

Consider three planes:
$P_1: x-y+z=1$
$P_2: x+y-z=-1$
$P_3: x-3y+3z=2$
Let $L_1, L_2, L_3$ be the lines of intersection of the planes $P_2$ and $P_3$,$P_3$ and $P_1$,and $P_1$ and $P_2$,respectively.
$STATEMENT-1$: At least two of the lines $L_1, L_2$ and $L_3$ are non-parallel.
$STATEMENT-2$: The three planes do not have a common point.

The equation of a plane parallel to the $x$-axis is:

If a plane is at a distance of $6$ units from the origin and the vector $2 \hat{i} + 6 \hat{j} - 3 \hat{k}$ is its normal,then the equation of the plane in Cartesian form is

The distance of the point $(1,3,-7)$ from the plane passing through the point $(1,-1,-1)$ having normal perpendicular to both the lines $\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-4}{3}$ and $\frac{x-2}{2}=\frac{y+1}{-1}=\frac{z+7}{-1}$ is