AS June 2025 Paper 1 Q16
16 A circle \(C\) is drawn on an Argand diagram.
The roots of the equation \(w^2 + 4w + 9 = 0\) lie on \(C\)
(a) Show that the roots of the equation\[w^2 + 4w + 9 = 0\]
are
\[-2 + \mathrm{i}\sqrt{5} \quad \text{and} \quad -2 - \mathrm{i}\sqrt{5}\] [2 marks](b) Explain briefly why the centre of \(C\) must lie on the real axis. [1 mark]
(c) The point \(7 + \mathrm{i}\sqrt{14}\) also lies on \(C\)
(i) Find the real number which represents the centre of \(C\) [2 marks]
(ii) Find the equation of \(C\)
Give your answer in the form \(|z - a| = b\) where \(a\) and \(b\) are constants. [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Completes the square to obtain \((w + 2)^2 = -5\) or Substitutes \(a = 1\), \(b = 4\), \(c = 9\) into the quadratic formula. or Calculates the sum and product of the roots. or Substitutes at least one of the given roots into the equation. or Expands \(\left(w - \left(-2 + \mathrm{i}\sqrt{5}\right)\right)\left(w - \left(-2 - \mathrm{i}\sqrt{5}\right)\right)\) or Rearranges the equation \(w = -2 \pm \mathrm{i}\sqrt{5}\) to obtain \((w + 2)^2 = -5\) | M1 | 1.1a |
| Completes a reasoned argument to prove the required result. Must include a conclusion which could be \(w^2 + 4w + 9 = 0\) or \(w = -2 \pm \mathrm{i}\sqrt{5}\) | R1 | 2.1 |
| (2) |
Typical solution
\[w^2 + 4w + 4 = -5\]\[(w + 2)^2 = -5\]\[w + 2 = \pm\sqrt{-5}\]\[w = -2 \pm \mathrm{i}\sqrt{5}\]| Scheme | Marks | AO |
|---|---|---|
| Explains why the centre lies on the real axis. Condone an incomplete explanation, eg the real axis is the perpendicular bisector of the chord. Accept an algebraic proof that the imaginary part is 0 | E1 | 2.4 |
| (1) |
Typical solution
Only points on the real axis are equidistant from \(-2 + \mathrm{i}\sqrt{5}\) and \(-2 - \mathrm{i}\sqrt{5}\)
| Scheme | Marks | AO |
|---|---|---|
| (i) Forms a correct equation in terms of the centre of \(C\) | M1 | 3.1a |
| Obtains 3 Accept \(3 + 0\mathrm{i}\) Condone \((3, 0)\) | A1 | 1.1b |
| (2) | ||
| (ii) Uses their centre to form an expression for the radius of \(C\) | M1 | 3.1a |
| Obtains a correct radius (excluding \(\sqrt{5}\) and \(\sqrt{14}\)) for their centre. FT their real centre to any one of the given points. | A1F | 1.1b |
| Obtains \(|z - 3| = \sqrt{30}\) FT their real centre and real radius. Note: M1A0A1 can be awarded. | A1F | 3.2a |
| (3) | ||
| (8 marks) |
Typical solution
(i)
Let centre of \(C\) be \(x\) where \(x \in \mathbf{R}\)
\[\sqrt{(x + 2)^2 + \sqrt{5}^{\,2}} = \sqrt{(7 - x)^2 + \sqrt{14}^{\,2}}\]\[x^2 + 4x + 9 = x^2 - 14x + 63\]\[18x = 54\]\[x = 3\](ii)
\[\begin{aligned}\text{radius} &= \sqrt{(7 - 3)^2 + \sqrt{14}^{\,2}} \\ &= \sqrt{30}\end{aligned}\]Locus of \(z\) is \(|z - 3| = \sqrt{30}\)