2d geometry system practice · Mixed difficulty

The locus of a point which moves in such a way that its distance from $(0,0)$ is three times its distance from the $x$ - axis, as given by

The triangle formed by $P(-2,3), Q(3,-2)$ and $R(0,0)$ is

The locus of a point $P$ which moves in such a way that the segment $O P$, where $O$ is the origin, has slope $\sqrt{3}$ is

The area of the pentagon, whose vertices are $A(1,1), B(7,21), C(7,-3), D(12,2)$ and $E(0,-3)$, is

Three distinct points $A, B$ and $C$ are given in the 2-dimensional coordinates plane such that the ratio of the distance of any one of them from the point $(1,0)$ to the distance from the point $(-1,0)$ is equal to $\frac{1}{3}$. Then, the circumcentre of the triangle $A B C$ is at the point

If vertices of a quadrilateral are $A(0,0), B(3,4), C(7,7)$ and $D(4,3)$, then quadrilateral $A B C D$ is

The centroid of a triangle lies at the origin and the coordinates of its two vertices are $(-8,7)$ and $(9,4)$. The area of the triangle is