FP2 June 2006 Q8
8. The point \(P\) represents a complex number \(z\) on an Argand diagram, where \[|z - 6 + 3\mathrm{i}| = 3|z + 2 - \mathrm{i}|.\]
(a) Show that the locus of \(P\) is a circle, giving the coordinates of the centre and the radius of this circle. (7)
The point \(Q\) represents a complex number \(z\) on an Argand diagram, where \[\tan\left[\arg(z + 6)\right] = \frac{1}{2}.\]
(b) On the same Argand diagram, sketch the locus of \(P\) and the locus of \(Q\). (5)
(c) On your diagram, shade the region which satisfies both \[|z - 6 + 3\mathrm{i}| \gt 3|z + 2 - \mathrm{i}| \ \text{ and } \ \tan\left[\arg(z + 6)\right] \gt \frac{1}{2}.\] (2)
| Scheme | Marks |
|---|---|
| Let \(z = x + \mathrm{i}y\) | |
| \((x - 6)^2 + (y + 3)^2 = 9[(x + 2)^2 + (y - 1)^2]\) | M1 |
| Leading to \(\ 8x^2 + 8y^2 + 48x - 24y = 0\) | M1 A1 |
| This is a circle; the coefficients of \(x^2\) and \(y^2\) are the same and there is no \(xy\) term. Allow equivalent arguments and ft their \(\mathrm{f}(x, y)\) if appropriate. | A1ft |
| \((x^2 + 6x + y^2 - 3y = 0)\) Leading to \((x + 3)^2 + \left(y - \tfrac{3}{2}\right)^2 = \tfrac{45}{4}\) | M1 |
| Centre: \(\left(-3, \tfrac{3}{2}\right)\) | A1 |
| Radius: \(\tfrac{3}{2}\sqrt{5}\) or equivalent | A1 |
| (7) |
Notes
Alternative
| Scheme | Marks |
|---|---|
| Accept the following argument:- The locus of \(P\) is a Circle of Apollonius, which is a circle with diameter \(XY\), where the points \(X\) and \(Y\) cut \((6, -3)\) and \((-2, 1)\) internally and externally in the ratio 3 : 1. | M1 A1 |
| \(X\): \((0, 0)\) \(Y\): \((-6, 3)\) | M1 A1 |
| Centre: \(\left(-3, \tfrac{3}{2}\right)\) | M1 A1 |
| Radius: \(\tfrac{3}{2}\sqrt{5}\) or equivalent | A1 |
| Scheme | Marks |
|---|---|
| Circle | B1 |
| centre in correct quadrant | B1 ft |
| through origin | B1 |
| Line cuts \(-\)ve \(x\) and \(+\)ve \(y\) axes | B1 |
| intersects with circle on axes and all correct | B1 |
| (5) |
| Scheme | Marks |
|---|---|
| Shading inside circle | B1 |
| and above line with all correct | B1 |
| (2) | |
| (14 marks) |