Find the reflection of the point $(-1, 3, 4)$ in the plane $x - 2y = 0$.

$A$ square $ABCD$ of diagonal $2a$ is folded along the diagonal $AC$ so that the planes $DAC$ and $BAC$ are at a right angle. The shortest distance between $DC$ and $AB$ is

If $A=(2,3,4)$ and $B=(-2,3,4)$,then the locus of a point $P(x,y,z)$ such that $PA+PB=4$ is

$A(27, -243, 81)$ is a point in space. $B, C, D$ are images of $A$ with respect to $XY, YZ$, and $ZX$ planes respectively. If the centroid of the triangle $BCD$ is $(\alpha, \beta, \gamma)$, then $\alpha + \beta + \gamma =$

Let $A(1, 2, 3)$, $B(-1, 4, 6)$, $C(0, -6, 4)$, and $D(1, 1, 1)$ be the vertices of a tetrahedron. Let $G$ be its centroid and $G_1$ be the centroid of its face $BCD$. Then $\frac{AG_1}{AG} =$

$A$ straight line drawn from the point $P(1,3,2)$,parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$,intersects the plane $L_1: x-y+3z=6$ at the point $Q$. Another straight line which passes through $Q$ and is perpendicular to the plane $L_1$ intersects the plane $L_2: 2x-y+z=-4$ at the point $R$. Then which of the following statements is(are) $TRUE$?
$(A)$ The length of the line segment $PQ$ is $\sqrt{6}$
$(B)$ The coordinates of $R$ are $(1,6,0)$
$(C)$ The centroid of the triangle $PQR$ is $\left(\frac{4}{3}, \frac{14}{3}, \frac{5}{3}\right)$
$(D)$ The perimeter of the triangle $PQR$ is $\sqrt{6}+\sqrt{13}+\sqrt{11}$