AS June 2020 Paper 1 Q18
18 The locus of points \(L_1\) satisfies the equation \(|z| = 2\)
The locus of points \(L_2\) satisfies the equation \(\arg(z + 4) = \dfrac{\pi}{4}\)
(a) Sketch \(L_1\) on the Argand diagram below. [1 mark]
(b) Sketch \(L_2\) on the Argand diagram above. [1 mark]
(c) The complex number \(a + \mathrm{i}b\), where \(a\) and \(b\) are real, lies on \(L_1\)
The complex number \(c + \mathrm{i}d\), where \(c\) and \(d\) are real, lies on \(L_2\)
Calculate the least possible value of the expression
\[(c - a)^2 + (d - b)^2\][3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Draws a circle with centre \((0, 0)\) and radius 2. Accept a reasonably accurate freehand circle. | B1 | 1.1b |
Typical solution
| Scheme | Marks | AO |
|---|---|---|
| Draws a straight line from \((-4, 0)\) at \(\frac{\pi}{4}\) to the real axis. Accept a reasonably accurate unruled line. | B1 | 1.1b |
Typical solution
| Scheme | Marks | AO |
|---|---|---|
| Selects a method to find the required expression by relating it to the shortest distance between the circle and the line. e.g. a perpendicular drawn from the line to the origin (or to the circle). or an indication of the use of the point \((-2, 2)\). | M1 | 3.1a |
| Calculates the distance from \((-2, 2)\) to the origin, or the distance from \((-2, 2)\) to \((-\sqrt{2}, \sqrt{2})\). | M1 | 1.1a |
| Obtains the correct value = \(12 - 8\sqrt{2}\) ACF, need not be simplified, exact value not required. | A1 | 3.2a |
| (5 marks) |
Typical solution