FP2 June 2010 Q6
6. A complex number \(z\) is represented by the point \(P\) in the Argand diagram.
(a) Given that \(|z - 6| = |z|\), sketch the locus of \(P\). (2)
(b) Find the complex numbers \(z\) which satisfy both \(|z - 6| = |z|\) and \(|z - 3 - 4\mathrm{i}| = 5\). (3)
The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \dfrac{30}{z}\).
(c) Show that \(T\) maps \(|z - 6| = |z|\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle. (5)
| Scheme | Marks |
|---|---|
| Vertical Straight line | B1 |
| Through 3 on real axis | B1 |
| (2) |
| Scheme | Marks |
|---|---|
| These are points where line \(x = 3\) meets the circle centre \((3, 4)\) with radius 5. | M1 |
| The complex numbers are \(3 + 9\mathrm{i}\) and \(3 - \mathrm{i}\). | A1 A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(|z - 6| = |z| \Rightarrow \left|\frac{30}{w} - 6\right| = \left|\frac{30}{w}\right|\) | M1 |
| \(\therefore |30 - 6w| = |30| \quad \Rightarrow \quad \therefore |5 - w| = |5|\) | M1 A1 |
| This is a circle with Cartesian equation \((u - 5)^2 + v^2 = 25\) | M1 A1 |
| (5) | |
| (10 marks) |