Let $\mathrm{C}$ be the centroid of the triangle with vertices $(3,-1),(1,3)$ and $(2,4) .$ Let $P$ be the point of intersection of the lines $x+3 y-1=0$ and $3 \mathrm{x}-\mathrm{y}+1=0 .$ Then the line passing through the points $\mathrm{C}$ and $\mathrm{P}$ also passes through the point

If in a parallelogram $ABDC$, the coordinates of $A, B$ and $C$ are respectively $(1, 2), (3, 4)$ and $(2, 5)$, then the equation of the diagonal $AD$ is

If the coordinates of the points $A,\, B,\, C$ be $(-1, 5),\, (0, 0)$ and $(2, 2)$ respectively and $D$ be the middle point of $BC$, then the equation of the perpendicular drawn from $B$ to the line $AD$ is

Given three points $P, Q, R$ with $P(5, 3)$ and $R$ lies on the $x-$ axis. If equation of $RQ$ is $x -2y = 2$ and $PQ$ is parallel to the $x-$ axis, then the centroid of $\Delta PQR$ lies on the line

If $A$ is $(2, 5)$, $B$ is $(4, -11)$ and $ C$ lies on $9x + 7y + 4 = 0$, then the locus of the centroid of the $\Delta ABC$ is a straight line parallel to the straight line is

The point moves such that the area of the triangle formed by it with the points $(1, 5)$ and $(3, -7)$ is $21$ sq. unit. The locus of the point is