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Not really sure where to start...hints are appreciated, thanks.

In this problem, we will show that the composition of two dilations is, in general, another dilation.

(a) Let $ z_0$ be an arbitrary complex number. We perform a dilation on $z_0,$ centered at $c_1$ with scale factor $k_1$, to get $z_1$. Express $z_1$ in terms of $z_0, c_1,$ and $k_1$.

(b) We then perform a dilation on $z_1$, centered at $c_2$ with scale factor $k_2$, to get $z_2.$ Express $z_2$ in terms of $z_0, c_1, c_2, k_1,$ and $k_2.$

(c) Show that if $k_1 k_2 \neq 1$, then there exists a single complex number c such that for any choice of $z_0$, the $z_2$ produced by parts (a) and (b) is a dilation of $z_0$ centered at this same value of c. Furthermore, show that c lies on the line joining $c_1$ and $c_2$.

Picture

(d) If $k_1 k_2 = 1$, then what is the transformation that takes $z_0$ to $z_2$?

Freedom
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1 Answers1

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Hint. Probably best to use "polar form not centred at the origin".

For (a), what is the relationship between $$|z_0-c_1|\quad\hbox{and}\quad |z_1-c_1|\ ?$$ Then, what is the relationship between $$\arg(z_0-c_1)\quad\hbox{and}\quad \arg(z_1-c_1)\ ?$$

Or perhaps even better, just draw a diagram indicating $c_1$, $z_0$ and $z_1$.

David
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