Edexcel IAL Mathematics FP2.3.4 z-plane and w-plane transformations

These questions are algebra-heavy: solve the transformation for z, substitute the original locus, then complete the square in u and v for the image.

Exam points

Rearrange fractional mappings such as w=(z−i)/(z+1) to express z in w.
Substitute line or circle loci from the z-plane into equations in u and v.
Complete the square to state the image circle's centre, radius or modulus form.

[Maximum number: 6]

A complex number z is represented by the point P in an Argand diagram.

Given that

|z-2 i|=|z-3|

find the equation of C, giving your answer in the form

|w-(p+q i)|=r

where p, q and r are real numbers to be determined.

$w=\frac{iz}{z-2i}\Rightarrow z=\frac{2wi}{w-i}$
Let w=u+iv. Substitution into the locus of z and simplification gives
$\left(u+\frac6{13}\right)^2+\left(v-\frac9{13}\right)^2=\frac4{13}.$
Therefore
$\left|w-\left(-\frac6{13}+\frac9{13}i\right)\right|=\frac{2}{\sqrt{13}}.$
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FP2.3.4 - Transformations between the z-plane and w-plane question 1