The equation of the locus of a point whose distance from $X Y$-plane is twice its distance from $Z$-axis is

If $\alpha$ is the angle between any two diagonals of a cube and $\beta$ is the angle between a diagonal of a cube and a diagonal of its face, which intersects this diagonal of the cube, then $\cos \alpha+\cos ^2 \beta=$

If the angle between the planes $a x-y+3 z=2 a$ and $3 x+a y+z=3 a$ is $\frac{\pi}{3}$, then the direction ratio of the line perpendicular to the plane $(a+2) x+(a-4) y+2 a z=a$ are

The number of values of ' $k$ ' for which the points $(-4,9, k),(-1,6, k),(0,7,10)$ from right-angled isosceles triangle is