A2 June 2022 Paper 1 Q8
8 Two sets of complex numbers are given by \(\left\{z : \arg(z - 10) = \tfrac{3}{4}\pi\right\}\) and \(\{z : |z - 3 - 6\mathrm{i}| = k\}\), where \(k\) is a positive constant. In an Argand diagram, one of the points of intersection of the two loci representing these sets lies on the imaginary axis.
Find the complex numbers represented by the points of intersection. [7]
| Scheme | Marks | AO |
|---|---|---|
 | M1 A1 B1 B1 | 1.1 1.1 1.1 1.1 |
| [4] |
Notes
M1: half line from 10
A1: at \(45^\circ\) to Real axis implied by angle shown or meeting the Imaginary axis at 10
B1: circle centre \(3 + 6i\)
B1: circle meeting half line on Imaginary axis
| Scheme | Marks | AO |
|---|---|---|
| DR one point is \(10\mathrm{i}\) | B1 | 1.1 |
| \(k^2 = (3 - 0)^2 + (6 - 10)^2\) | M1* | 3.1a |
| \(\Rightarrow k = 5\) | A1 | 1.1 |
| line \(x + y = 10\) \((x - 3)^2 + (10 - x - 6)^2 = 25\) | M1dep* | 3.1a |
| \(\Rightarrow 2x^2 - 14x = 0\) | M1 | 1.1 |
| \(\Rightarrow x = 7,\ y = 3\) | A1 | 1.1 |
| other point is \(7 + 3i\) | A1 | 3.2a |
| [7] |
Notes
A1: soi
M1dep*: solving \(x + y = 10\) and circle equation simultaneously
M1: Rearranging into a quadratic = 0
A1: Could see solutions as \(\sqrt{58}(\cos 0.405 + i\sin 0.405)\) or \(\sqrt{58}e^{0.405i}\)
Alternative solution
| Scheme | Marks |
|---|---|
| one point is \(10\mathrm{i}\) | B1 |
| eqn of perp from (3, 6) to chord is \(y = x + 3\) | M1 |
| solving with \(x + y = 10\) midpoint of chord is \(\left(3\frac{1}{2},\ 6\frac{1}{2}\right)\) | M1 A1 |
| other end of chord is \(\left(2 \times 3\frac{1}{2} - 0,\ 2 \times 6\frac{1}{2} - 10\right)\) | M1 |
| \(\Rightarrow\) other point of intersection is (7, 3) | A1 |
| this represents \(7 + 3i\) | A1 |
| [7] |
M1: oe, e.g. midpoint of chord has equal \(x\) and \(y\) displacements from (3, 6) or by inspection
A1: oe
A1: Could see solutions as \(\sqrt{58}(\cos 0.405 + i\sin 0.405)\) or \(\sqrt{58}e^{0.405i}\)