The length of the foot of the perpendicular from the point $P\left(1, \frac{3}{2}, 2\right)$ to the plane $2x - 2y + 4z + 17 = 0$ is

$A$ plane passing through the points $A(2, 3, 5)$ and $B(-3, -5, -7)$ is perpendicular to the plane $x - y + z = 1$. Which of the following points lies on this plane?

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

If the plane $2x + 3y + 5z = 1$ intersects the coordinate axes at the points $A, B, C$,then the centroid of $\triangle ABC$ is

Find the equation of a plane,given that the foot of the perpendicular drawn to the plane from the origin is $(2, 1, 2)$.

$A$ plane $\Pi$ passes through the points $A=(0,0,2)$,$B=(1,0,1)$,and $C=(3,1,1)$. If the plane $\Pi$ makes angles $\alpha$ and $\beta$ with the $XY$ and $XZ$-coordinate planes respectively,then $\sin^2 \alpha + \sin^2 \beta =$