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

Let two planes be $P_1: 2x - y + z = 2$ and $P_2: x + 2y - z = 3$. Find the equation of the plane passing through the point $(-1, 3, 2)$ and perpendicular to both planes $P_1$ and $P_2$.

If the distance between the plane,$23x - 10y - 2z + 48 = 0$ and the plane containing the lines $\frac{x+1}{2} = \frac{y-3}{4} = \frac{z+1}{3}$ and $\frac{x+3}{2} = \frac{y+2}{6} = \frac{z-1}{\lambda}$ $(\lambda \in R)$ is equal to $\frac{k}{\sqrt{633}}$,then $k$ is equal to

If $A(2,1,-1)$,$B(6,-3,2)$,and $C(-3,12,4)$ are the vertices of a triangle $ABC$,and the equation of the plane containing the triangle $ABC$ is $53x + by + cz + d = 0$,then find the value of $\frac{d}{b+c}$.

For scalars $\lambda, \mu$,if the vector equation of a plane is $r=(2+3 \lambda-\mu) \hat{i}+(1-2 \lambda+3 \mu) \hat{j}+(-2+2 \lambda+\mu) \hat{k}$,then its Cartesian equation is

If $O$ is the origin and the coordinates of $P$ are $(1, 2, -3),$ find the equation of the plane passing through $P$ and perpendicular to $OP.$