Consider the curve with equation\({Re}\left( \dfrac{1}{z} \right) = \dfrac{1}{6}.\)
For each complex number in the following list,\(1, 4i, 3+3i, 3-3i, 1 - 2i, 2+ 3i, 6,\)
figure out whether each one is on the curve, then enter "yes" or "no" in the blank corresponding to each option below.
+6248 \(1: \text{ obviously no}\\ 4i: \text{ hopefully obviously no}\\ 3+3i:~Re\left(\dfrac{1}{3+3i}\right) = Re\left(\dfrac{3-3i}{18}\right) = \dfrac 1 6 \text{ so yes}\\ 3-3i: \text{same answer as for }3+3i\\ 1-2i: Re\left(\dfrac{1+2i}{5}\right) = \dfrac 1 5 \neq \dfrac 1 6 \text{ so no}\\ 2+3i: Re\left(\dfrac{2-3i}{13}\right) = \dfrac{2}{13} \neq \dfrac 1 6 \text{ so no}\\ 6: \text{ obviously yes}\)
.+6248 \(1: \text{ obviously no}\\ 4i: \text{ hopefully obviously no}\\ 3+3i:~Re\left(\dfrac{1}{3+3i}\right) = Re\left(\dfrac{3-3i}{18}\right) = \dfrac 1 6 \text{ so yes}\\ 3-3i: \text{same answer as for }3+3i\\ 1-2i: Re\left(\dfrac{1+2i}{5}\right) = \dfrac 1 5 \neq \dfrac 1 6 \text{ so no}\\ 2+3i: Re\left(\dfrac{2-3i}{13}\right) = \dfrac{2}{13} \neq \dfrac 1 6 \text{ so no}\\ 6: \text{ obviously yes}\)
