Wednesday, 9 November 2011

A complex number problem

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 $\frac{\pi}{3}$

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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