The vector equation of the plane passing through the line of intersection of the planes $x+y+z=1$ and $2x+3y+4z=5$,which is perpendicular to the plane $x-y+z=0$,is

If the product of distances of the point $(1, 1, 1)$ from the origin and the plane $x - y + z + k = 0$ is $5$,then $k =$

The $xy$-plane divides the line segment joining the points $(-1, 3, 4)$ and $(2, -5, 6)$ in which ratio?

In $R^3$,consider the planes $P_1: y=0$ and $P_2: x+z=1$. Let $P_3$ be a plane,different from $P_1$ and $P_2$,which passes through the intersection of $P_1$ and $P_2$. If the distance of the point $(0,1,0)$ from $P_3$ is $1$ and the distance of a point $(\alpha, \beta, \gamma)$ from $P_3$ is $2$,then which of the following relations is (are) true?
$(A)$ $2\alpha+\beta+2\gamma+2=0$
$(B)$ $2\alpha-\beta+2\gamma+4=0$
$(C)$ $2\alpha+\beta-2\gamma-10=0$
$(D)$ $2\alpha-\beta+2\gamma-8=0$

Let $P$ be a point in the first octant,whose image $Q$ in the plane $x+y=3$ (that is,the line segment $PQ$ is perpendicular to the plane $x+y=3$ and the mid-point of $PQ$ lies in the plane $x+y=3$) lies on the $z$-axis. Let the distance of $P$ from the $x$-axis be $5$. If $R$ is the image of $P$ in the $xy$-plane,then the length of $PR$ is.

The equation of the plane containing the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$ is