1993 Paper 3 Q6

Year: 1993
Paper: 3
Question Number: 6

Course: LFM Stats And Pure
Section: Complex Numbers (L8th)

Difficulty: 1700.0 Banger: 1484.0

complex numbers Argand diagram circle locus conjugate Mobius transformation geometric proof

Problem

The point in the Argand diagram representing the complex number $z$ lies on the circle with centre $K$ and radius $r$, where $K$ represents the complex number $k$. Show that $$ zz^* -kz^* -k^*z +kk^* -r^2 =0. $$ The points $P$, $Q_1$ and $Q_2$ represent the complex numbers $z$, $w_1$ and $w_2$ respectively. The point $P$ lies on the circle with $OA$ as diameter, where $O$ and $A$ represent $0$ and $2i$ respectively. Given that $w_1=z/(z-1)$, find the equation of the locus $L$ of $Q_1$ in terms of $w_1$ and describe the geometrical form of $L$. Given that $w_2=z^*$, show that the locus of $Q_2$ is also $L$. Determine the positions of $P$ for which $Q_1$ coincides with $Q_2$.

No solution available for this problem.

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Difficulty Rating: 1700.0

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Banger Rating: 1484.0

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Problem source

The point in the Argand diagram representing the complex number
$z$ lies on the circle with centre $K$ and radius $r$, where $K$
represents the complex number $k$. Show that 
$$
zz^* -kz^* -k^*z +kk^* -r^2 =0.
$$
The points $P$, $Q_1$ and $Q_2$ represent the complex numbers
$z$, $w_1$ and $w_2$ respectively. The point $P$ lies on the circle
with $OA$ as diameter, where $O$ and $A$ represent $0$ and 
$2i$ respectively. Given that $w_1=z/(z-1)$, find the equation of the 
locus $L$ of $Q_1$ in terms of $w_1$ and describe 
the geometrical form of $L$.
Given that $w_2=z^*$, show that the locus of $Q_2$ is also $L$. Determine the
positions of $P$ for which $Q_1$ coincides with $Q_2$.

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