Challenging +1.2 This is a Möbius transformation question requiring students to find the image of a circle. While it involves Further Maths content (complex transformations), the method is standard: substitute z = 2e^(iθ), manipulate to find |w - a| = r form, or use three points. It's more involved than routine A-level questions but follows a well-established technique taught in F2, making it moderately challenging rather than requiring novel insight.
4. A transformation from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { z } { z + 3 } , \quad z \neq - 3$$
Under this transformation, the circle \(| z | = 2\) in the \(z\)-plane is mapped onto a circle \(C\) in the \(w\)-plane.
Determine the centre and the radius of the circle \(C\).
Rearrange to suitable form for a circle and attempt centre and/or radius; \(-16/25\) may be omitted
Centre \(\left(-\frac{4}{5}, 0\right)\)
A1
oe
Radius \(\frac{6}{5}\)
A1
oe
# Question 4:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $w = \frac{z}{z+3} \Rightarrow z = \frac{3w}{1-w}$ | M1A1 | M1: Attempt to make $z$ the subject; A1: Correct expression for $z$ |
| $|z|=2 \Rightarrow \left|\frac{3w}{1-w}\right|=2$, $|3w|=2|1-w|$, $9(u^2+v^2)=4(u-1)^2+4v^2$ | M1A1 | M1: Uses $|z|=2$ to obtain equation in $u$ and $v$; Pythagoras must be used correctly, no $i$ seen; A1: Any correct equation in $u$ and $v$ |
| $5u^2+5v^2+8u-4=0$ | — | — |
| $\left(u+\frac{4}{5}\right)^2+v^2-\frac{16}{25}-\frac{4}{5}=0$ | M1 | Rearrange to suitable form for a circle and attempt centre and/or radius; $-16/25$ may be omitted |
| Centre $\left(-\frac{4}{5}, 0\right)$ | A1 | oe |
| Radius $\frac{6}{5}$ | A1 | oe |
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4. A transformation from the $z$-plane to the $w$-plane is given by
$$w = \frac { z } { z + 3 } , \quad z \neq - 3$$
Under this transformation, the circle $| z | = 2$ in the $z$-plane is mapped onto a circle $C$ in the $w$-plane.
Determine the centre and the radius of the circle $C$.\\
\hfill \mbox{\textit{Edexcel F2 2014 Q4 [7]}}