1) Show that $Re(z)^2-Re(z^2)\geq 0$

2) Write down the section formula and linear representation of a line.3) Write down the equation of line on argand plane equivalent to y=x.

4) (*) Write down the equation of a hyperbola, like xy=1, on argand diagram.

Now in the above diagram, from left to right, gives three parallel stright lines, namely $l_1,l_2,l_3$. The small circle is a unit circle and $C_i:|z|=r_i\in R$ is the circle with $l_i$ as tangent. L is the line passing through origin perpendicular to $l_i$ while $u_i$ are points of intersection between $L,l_i$ .

Given that:

1) $l_2$ is represented by $|z-u_3|=|z-u_1|$

2) $Re(u_2)=r_1$.

3) The inclination of $l_i$ is $\frac{\pi}{3}$

Problem:

Basic problem

a) Show that $u_2=\frac{1}{2}(u_1+u_3)$

b) Show that $u_i=l_i(cis\theta)$ where $\theta \in R$.

c) By (a), (b) or otherwise, write down the equation/values of $u_i,l_i,r_i,L$ in numerical form.

Advanced problem:

d) Two tangent of $C_1$ passing through $u_3$ touch $C_1$ at two points, a and b respectively. Show that line passing through a and b is parallel to $l_1$.

e) i) Let the intersection point between $Re(z)=r_1$ and $l_1$ be $v_1$. Find the equation of circle $C_v$ if $0,u_1,v_1,r_1$ lies on $C_v$.

ii) Is $Re(z_3)=r_2$? Prove your assertion.

Extreme problem:

f) A function $f(x)=z$ is defined by:

Step I: w is a point on $l_1$ outside of $C_1$ such that $|u_1-w|=x$.

Step II: $l_w$ is the tangent from $w$ to $C_1$ and touch $C_1$ at $w'$.

Step III: z is the point on $l_w$, not lying between w and w' and $|z-w'|=1$.

i) Write down domain and codomain of f.

ii) Show that f is injective.

iii) Sketch f on the argand diagram.

g) Another function $g(x)=z$ is similar to f, but this time in step III, z is a point that $z=\frac{1}{2}(w+w')$.

i) Show that g is also injective.

ii) Find the area enclosed by $l_w,C_1,L$.

iii) Sketch g.

iv) What's the difference in nature between shape of f and g?

v) Sketch $h(x)=g(x)-f(x)$.

h) If $u_2$ is now replaced by $u_3$ in determining new f and g, say a(x) and b(x), is a and b a linear transform of f and g? Explain.

Pretty hard this time. Can these questions be easily solved in Cartesian plane? [A technique of rotating axis is required]

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