6. Four of these points lie on a circle. Which of the points does not lie on that circle?
A $(5,0)$
B $(4,3)$
C $(2,2)$
D $(3,4)$
$\mathrm{E}(0,5)$
6. C The distance, $d$, of the point $(x, y)$ from the origin $O,(0,0)$, is given by $d=\sqrt{x^{2}+y^{2}}$. It is clear that the points $(5,0),(4,3),(3,4),(0,5)$ are all at a distance of 5 units from $O$ as $\sqrt{4^{2}+3^{2}}=\sqrt{3^{2}+4^{2}}=5$. So those four points all lie on a circle with centre $O$ and radius 5 .
However, the distance from the point $(2,2)$ to $O$ is $\sqrt{2^{2}+2^{2}}=\sqrt{8} \neq 5$.
Therefore the point $(2,2)$ does not lie on the same circle as the other four points.
(There is a unique circle through any three non-collinear points, so there can be no circle that passes through the point $(2,2)$ and three of the other four given points.)