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Two complex numbers $z_1$ and $z_2$ satisfy $|z_2|=1$ and $\dfrac{z_1-2z_2}{2-z_1\bar{z}_2}$ is unimodular, but $z_2$ is not unimodular. Where does $z_1$ lie?
Let $w = \frac{z_1-2z_2}{2-z_1\bar{z}_2}$, with $|w|=1$.
$|z_1-2z_2|^2 = |2-z_1\bar{z}_2|^2$
$(z_1-2z_2)\overline{(z_1-2z_2)} = (2-z_1\bar{z}_2)\overline{(2-z_1\bar{z}_2)}$
$|z_1|^2 - 2z_1\bar{z}_2 - 2\bar{z}_1z_2 + 4|z_2|^2 = 4 - 2z_1\bar{z}_2 - 2\bar{z}_1z_2 + |z_1|^2|z_2|^2$
$|z_1|^2 + 4|z_2|^2 = 4 + |z_1|^2|z_2|^2$
$|z_1|^2(1-|z_2|^2) = 4(1-|z_2|^2)$
Since $|z_2|\neq1$: $|z_1|^2=4\Rightarrow|z_1|=2$. So $z_1$ lies on a circle of radius 2.
A projectile has initial velocity $(2\hat{i}+3\hat{j})\text{ m/s}$ at point A (at ground level). At point B (on the x-axis, same height as A), its velocity is:
Horizontal component is unchanged throughout: $v_x = 2\text{ m/s}$.
When the projectile returns to the same height, the vertical component reverses: $v_y = -3\text{ m/s}$.
Velocity at B $= \mathbf{2\hat{i} - 3\hat{j}}$
Thinking about the relationship:
- An arch is the key structural element in a (stone/masonry) bridge — it defines and enables the bridge's spanning ability.
- A dome is the key structural/spatial element that defines a rotunda — a circular room covered by a dome.
A rotunda is specifically a round building or room topped with a dome. Famous examples: Pantheon (Rome), US Capitol Rotunda.
A pendentive is the transitional element between a square base and a dome — not the completed building type. A vault is a different structural form. A minaret is a tower.
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