If a line makes angles $\alpha, \beta, \gamma, \delta$ with the four diagonals of a cube,then find the value of $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma + \sin^2 \delta$.

$A$ point moves such that the sum of its distances from $(4, 0, 0)$ and $(-4, 0, 0)$ is always $10$. The locus of the point is:

$A$ line with direction ratios $2, 1, 2$ intersects the lines $x = y + a = z$ and $x + a = 2y = 2z$. Find the coordinates of the points of intersection.

For $a \neq 0$,if the sum of the distances of a point $P(x, y, z)$ from the points $F_1(a, 0, 0)$ and $F_2(-a, 0, 0)$ is a constant $2k$,then the locus of that point is

Find the equation of the set of points $P$,the sum of whose distances from $A(4, 0, 0)$ and $B(-4, 0, 0)$ is equal to $10$.

$\Pi_1, \Pi_2, \Pi_3$ are three planes which are respectively parallel to the $YZ, ZX$ and $XY$ planes at distances $a, b$ and $c$ forming a rectangular parallelopiped. $d_1$ is a diagonal of the face of $XY$-plane not passing through the origin and $d_2$ is a diagonal of the plane $\Pi_2$ coterminous with $d_1$. If none of the coordinates of the vertices of the parallelopiped are negative, then the angle between $d_1$ and $d_2$ is