1. ## Argand Plane?

If $\displaystyle \sum_{r = 1}^{4} a_{r} = 0$ and $\displaystyle \sum_{r = 1}^{4} a_{r}Z_{r} = 0$ where $\displaystyle a_{1},a_{2},a_{3},a_{4}$ are non-zero real numbers and $\displaystyle P(Z_{1}),Q(Z_{2}),R(Z_{3}), S(Z_{4})$ are concyclic points on Argand Plane, then prove that:
$\displaystyle \sum_{r = 1}^{4} a_{r}|Z_{r}|^2 = 0.48$

2. Originally Posted by fardeen_gen
If $\displaystyle \sum_{r = 1}^{4} a_{r} = 0$ and $\displaystyle \sum_{r = 1}^{4} a_{r}Z_{r} = 0$ where $\displaystyle a_{1},a_{2},a_{3},a_{4}$ are non-zero real numbers and $\displaystyle P(Z_{1}),Q(Z_{2}),R(Z_{3}), S(Z_{4})$ are concyclic points on Argand Plane, then prove that:
$\displaystyle \sum_{r = 1}^{4} a_{r}|Z_{r}|^2 = 0.48$
This cannot be correct. Suppose that $\displaystyle a_{1},a_{2},a_{3},a_{4}$ and $\displaystyle Z_{1},Z_{2},Z_{3},Z_{4}$ satisfy these conditions. If we replace $\displaystyle a_{1},a_{2},a_{3},a_{4}$ by $\displaystyle 2a_{1},2a_{2},2a_{3},2a_{4}$ then the conditions will still be satisfied, but $\displaystyle \sum_{r = 1}^{4} a_{r}|Z_{r}|^2$ will be doubled.

3. The problem set of the complex numbers section of my text seems to be full of wrong problems. This makes it two wrong problems in a day