Year: 1994
Paper: 3
Question Number: 6
Course: LFM Stats And Pure
Section: Complex Numbers (L8th)
No solution available for this problem.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1501.5
Banger Comparisons: 2
The four points $A,B,C,D$ in the Argand diagram (complex plane) correspond
to the complex numbers $a,b,c,d$ respectively. The point $P_{1}$
is mapped to $P_{2}$ by rotating about $A$ through $\pi/2$ radians.
Then $P_{2}$ is mapped to $P_{3}$ by rotating about $B$ through
$\pi/2$ radians, $P_{3}$ is mapped to $P_{4}$ by rotating about
$C$ through $\pi/2$ radians and $P_{4}$ is mapped to $P_{5}$ by
rotating about $D$ through $\pi/2$ radians, each rotation being
in the positive sense. If $z_{i}$ is the complex number corresponding
to $P_{i},$ find $z_{5}$ in terms of $a,b,c,d$ and $z_{1}.$
Show that $P_{5}$ will coincide with $P_{1},$ irrespective of the
choice of the latter if, and only if \[a-c=\mathrm{i}(b-d)\] and interpret
this condition geometrically.
The points $A,B$ and $C$ are now chosen to be distinct points on
the unit circle and the angle of rotation is changed to $\theta,$
where $\theta\neq0,$ on each occasion. Find the necessary and sufficient
condition on $\theta$ and the points $A,B$ and $C$ for $P_{4}$
always to coincide with $P_{1}.$