How do you find the image of a locus?

Express x,y (or z) in terms of u,v (or w), substitute into the locus equation, and simplify to an equation in u and v.

What does w=az+b do to a locus?

Enlargement by |a|, rotation through \arg a, then translation by b. Circles go to circles, lines to lines.

Image of |z|=2 under w=3z+2i?

The circle with centre 2i and radius 6.

What happens to modulus and argument under w=z^2?

The modulus is squared and the argument is doubled.

Write x in terms of u,v for w=\frac1z.

x=\frac{u}{u^2+v^2} (and y=\frac{-v}{u^2+v^2}).

Image of the line x=1 under w=\frac1z?

The circle u^2+v^2=u, centre \left(\frac12,0\right), radius \frac12.

Image of |z-1|=1 under w=\frac1z?

The line u=\frac12, because the circle passes through z=0.

Transformations from the z-plane to the w-planeEdexcel A-Level Further Maths: Flashcards

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What does the transformation $w=f(z)$ do?

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What does the transformation $w=f(z)$ do?: It maps each point $z=x+iy$ in the $z$ -plane to a point $w=u+iv$ in the $w$ -plane.
How do you find the image of a locus?: Express $x,y$ (or $z$ ) in terms of $u,v$ (or $w$ ), substitute into the locus equation, and simplify to an equation in $u$ and $v$ .
What does $w=az+b$ do to a locus?: Enlargement by $|a|$ , rotation through $\arg a$ , then translation by $b$ . Circles go to circles, lines to lines.
Image of $|z|=2$ under $w=3z+2i$ ?: The circle with centre $2i$ and radius $6$ .
For $w=z^2$ , give $u$ and $v$ .: $u=x^2-y^2$ and $v=2xy$ .
What happens to modulus and argument under $w=z^2$ ?: The modulus is squared and the argument is doubled.
Image of $|z|=3$ under $w=z^2$ ?: The circle $|w|=9$ .
Image of the half-line $\arg z=\alpha$ under $w=z^2$ ?: The half-line $\arg w=2\alpha$ .
Why is $w=z^2$ two-to-one?: $z$ and $-z$ have the same square.
Make $z$ the subject of $w=\frac{az+b}{cz+d}$ .: $z=\frac{b-dw}{cw-a}$ .
Write $x$ in terms of $u,v$ for $w=\frac1z$ .: $x=\frac{u}{u^2+v^2}$ (and $y=\frac{-v}{u^2+v^2}$ ).
Image of the line $x=1$ under $w=\frac1z$ ?: The circle $u^2+v^2=u$ , centre $\left(\frac12,0\right)$ , radius $\frac12$ .
Image of $|z-1|=1$ under $w=\frac1z$ ?: The line $u=\frac12$ , because the circle passes through $z=0$ .

Exam questions on Transformations from the z-plane to the w-plane

The transformation $T$ from the $z$ -plane to the $w$ -plane is given by $w=3z+2i$ , where $z=x+iy$ and $w=u+iv$ .
Find the Cartesian equation of the image of the line $y=x$ in the $z$ -plane.2 marks
The transformation $T$ from the $z$ -plane to the $w$ -plane is given by $w=z^2$ , where $z=x+iy$ and $w=u+iv$ .
Find the image of the line $y=x$ in the $w$ -plane.2 marks
The transformation $T$ from the $z$ -plane to the $w$ -plane is given by $w=\frac{1}{z}$ , where $z=x+iy$ and $w=u+iv$ .
Show that the image of the line $x=1$ in the $z$ -plane is a circle in the $w$ -plane, and find its centre and radius.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).