On the Argand diagram, $P$ represents the complex number $z$, and $R$ the number $\frac{1}{z}$ A square $PQRS$ is drawn in the plane with $PR$ as a diagonal If $P$ lies on the circle $|z| = 2$,
(I) Prove that $Q$ will lie on the ellipse whose equation has the form $\frac{x ^ 2}{a ^ 2} + \frac{y ^ 2}{b ^ 2} = 1$;
(II) Hence, specify the numerical values for $a$ and $b$.
Building upon your start:
The difference between $P$ and the midpoint is: $\frac{3}{4}\cos t+\frac{5}{4}i\sin t$
If we multiple this by $i$ we get a complex number which has been rotated by $90^\circ$. That complex number is: $-\frac{5}{4}\sin t+\frac{3}{4}i\cos t$
Next add this to the midpoint to get $Q$: $\frac{5}{4}\cos t-\frac{5}{4}\sin t+\frac{3}{4}i\cos t+\frac{3}{4}i\sin t$
$$=\frac{5}{4}(\cos t-\sin t)+\frac{3}{4}i(\cos t+\sin t)$$
$$=\frac{5\sqrt{2}}{4}\cos\left(t+\frac{\pi}{4}\right)+\frac{3\sqrt{2}}{4}i\sin\left(t+\frac{\pi}{4}\right)$$
You should recognise this as the parametric form of an ellipse with semi-major radius of $\frac{5\sqrt{2}}{4}$ and semi-minor radius of $\frac{3\sqrt{2}}{4}$.
Hence $a=\frac{5\sqrt{2}}{4}$ and $b=\frac{3\sqrt{2}}{4}$.
Note we could have multiplied by $-i$ for the $90^\circ$ rotation. This would have followed very similar steps and arrived at $Q$:
$$=\frac{5\sqrt{2}}{4}\cos\left(t-\frac{\pi}{4}\right)+\frac{3\sqrt{2}}{4}i\sin\left(t-\frac{\pi}{4}\right)$$
Which would lead to the same $a$ and $b$.
The complex number R is 0.5 * cos (-t) + 0.5 * i * sin (-t) for any t. This is the behavior of complex numbers with respect to their inverses.
The midpoint of the diagonal PR is 1.25 * cos (t) + 0.75 * i * sin (t) for any t.
The diagonal QS would have the same midpoint and length as PR, and QS ^ PR.
– Felix Zhu Jul 29 '16 at 02:54