Let $A B C$ be a triangle. Let a point $P$ divide $A B$ in the ratio $1: 2$ internally and a point $Q$ divide $B C$ in the ratio $1: 2$ internally. Let $D$ be the point of intersection of $A Q$ and $C P$. If the area of the $\triangle A B C$ is $k$ square units, then the area of the $\triangle B C D$ in (sq. units) is

$B(2,3), C(5,-2), D(1,-1)$ are three points. If $A$ is a variable point such that the area of the quadrilateral $A B C D$ is 10 sq. units, then the locus of $A$ is

A line makes intercepts 5 and 7 on the coordinate axes. The axes are rotated through an angle $\theta$ in the positive direction about the origin so that the line makes equal intercepts on the new axes, then $|\tan \theta|=$

$L \equiv 7 x-y+8=0$ is one of the diagonals of a square for which $(-4,5)$ and $(3,4)$ are two vertices. Then, the coordinates of the two vertices lying on the diagonal $L=0$ are