FP2 June 2008 Q10
10. The point \(P\) represents a complex number \(z\) on an Argand diagram such that \[|z - 3| = 2|z|.\]
(a) Show that, as \(z\) varies, the locus of \(P\) is a circle, and give the coordinates of the centre and the radius of the circle. (5)
The point \(Q\) represents a complex number \(z\) on an Argand diagram such that \[|z + 3| = |z - \mathrm{i}\sqrt{3}|.\]
(b) Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies. (5)
(c) On your diagram shade the region which satisfies \[|z - 3| \geqslant 2|z| \text{ and } |z + 3| \geqslant |z - \mathrm{i}\sqrt{3}|.\] (2)
| Scheme | Marks |
|---|---|
| \(|(x - 3) + \mathrm{i}y| = 2|x + \mathrm{i}y| \Rightarrow (x - 3)^2 + y^2 = 4x^2 + 4y^2\) | M1A1 |
| \(\therefore x^2 + y^2 + 2x - 3 = 0\) | |
| \((x + 1)^2 + y^2 = 4\) | M1 |
| Centre \((-1, 0)\), radius 2 | A1, A1 |
| (5) |
Notes
1st M: Use \(z = x + \mathrm{i}y\), and attempt square of modulus of each side.
Not squaring the 2 on the RHS would be M1 A0.
2nd M: Attempting to express in the form \((x - a)^2 + (y - b)^2 = k\), or attempting centre and radius from the form \(x^2 + y^2 + 2gx + 2fy + c = 0\)
| Scheme | Marks |
|---|---|
| Circle, centre on \(x\)-axis | B1 |
| \(C\ (-1, 0),\ r = 2\) Follow through centre and radius, but dependent on first B1. There must be indication of their ‘−3’, ‘−1’ or ‘1’ on the \(x\)-axis and no contradictory evidence for their radius. | dB1ft |
| Straight line | B1 |
| Straight line through \((-1, 0)\), or perp. bisector of \((-3, 0)\) and \((0, \sqrt{3})\). | B1 |
| Straight line through point of int. of circle & −ve \(y\)-axis, or through \((0, -\sqrt{3})\) | B1 |
| (5) |
| Scheme | Marks |
|---|---|
| Shading (only) inside circle | B1 |
| Inside correct circle and all of the correct side of the correct line… this mark is dependent on all the previous B marks in parts (b) and (c). | dB1 |
| (2) | |
| (12 marks) |