Year: 2005
Paper: 3
Question Number: 8
Course: LFM Stats And Pure
Section: Complex Numbers (L8th)
No solution available for this problem.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
In this question, $a$ and $c$ are distinct non-zero complex numbers.
The complex conjugate of any complex number $z$ is denoted by
$z^*$.
Show that
\[
|a - c|^2 = aa^* + cc^* -ac^* - ca^*
\]
and hence prove that the triangle
$OAC$ in the Argand diagram,
whose vertices are represented by
$0$, $a$ and $c$ respectively, is right angled at $A$
if and only if $2aa^* = ac^*+ca^*\,$.
Points $P$ and $P'$ in the Argand diagram
are represented by the complex numbers $ab$ and
$\ds \frac{a}{b^*}\,$, where $b$ is a non-zero complex number.
A circle in the Argand diagram has centre $C$ and passes through the point $A$,
and is such that $OA$ is a tangent to the circle.
Show that the point $P$ lies on the circle
if and only if the point $P'$ lies on the circle.
Conversely, show that if the points represented
by the complex numbers $ab$ and $\ds \frac{a}{b^*}$,
for some non-zero complex number $b$ with $bb^* \ne 1\,$,
both lie on a circle centre $C$ in the Argand diagram
which passes through $A$, then $OA$ is a tangent to the circle.