A plane $(\pi)$ passing through the point $(1,2,-3)$ is perpendicular to the planes $x+y-z+4=0$ and $2 x-y+z+1=0$. If the equation of the plane $(\pi)$ is $a x+b y+c z+1=0$, then $a^2+b^2+c^2=$

If the ratio of the perpendicular distances of a variable point $P(x, y, z)$ from the $X$-axis and from the $Y Z$ - plane is $2: 3$, then the equation of the locus of $P$ is

The direction cosines of two lines are connected by the relations $l-m+n=0$ and $2 l m-3 m n+n l=0$. If $\theta$ is the angle between these two lines, then $\cos \theta=$

A plane $\pi$ passes through the points $(5,1,2),(3,-4,6)$ and $(7,0,-1)$. If $p$ is the perpendicular distance from the origin to the plane $\pi$ and $l, m$ and $n$ are the direction cosines of a normal to the plane $\pi$, the $|3 l+2 m+5 n|=$