June 2022 Paper 1 Q3
3. A circle has equation
\[x^2 + y^2 - 10x + 16y = 80\]Given that \(P\) is the point on the circle that is furthest away from the origin \(O\),
| Scheme | Marks | AO |
|---|---|---|
| (i) \(x^2 + y^2 - 10x + 16y = 80 \Rightarrow (x - 5)^2 + (y + 8)^2 = \ldots\) | M1 | 1.1b |
| Centre \((5,\ -8)\) | A1 | 1.1b |
| (ii) Radius 13 | A1 | 1.1b |
| (3) |
Notes
(a)(i)
M1: Attempts to complete the square on both \(x\) and \(y\) terms.
Accept \((x \pm 5)^2 + (y \pm 8)^2 = \ldots\) or imply this mark for a centre of \((\pm 5,\ \pm 8)\)
Condone \((x \pm 5)^2 \ldots\ldots (y \pm 8)^2 = \ldots\) where the first … could be , or even \(-\)
A1: Correct centre \((5,\ -8)\).
Accept without brackets. May be written \(x = 5,\ y = -8\)
(a)(ii)
A1: 13. The M mark must have been awarded, so it can be scored following a centre of \((\pm 5,\ \pm 8)\).
Do not allow for \(\sqrt{169}\) or \(\pm 13\)
| Scheme | Marks | AO |
|---|---|---|
| Attempts \(\sqrt{\text{``}5\text{''}^2 + \text{``}8\text{''}^2} + \text{``}13\text{''}\) | M1 | 3.1a |
| \(13 + \sqrt{89}\) but ft on their centre and radius | A1ft | 1.1b |
| (2) | ||
| (5 marks) |
Notes
M1: Attempts \(\sqrt{\text{``}5\text{''}^2 + \text{``}8\text{''}^2} + \text{``}13\text{''}\) for their centre \((5,\ -8)\) and their radius 13.
Award when this is given as a decimal, e.g. 22.4 for correct centre and radius. Look for \(\sqrt{a^2 + b^2} + r\) where centre is \((\pm a,\ \pm b)\) and radius is \(r\)
A1ft: \(13 + \sqrt{89}\) Follow through on their \((5,\ -8)\) and their 13 leading to an exact answer. ISW for example if they write \(13 + \sqrt{89} = 22.4\)
There are more complicated attempts which could involve finding \(P\) by solving \(y = \text{``}{-}\dfrac{8}{5}x\text{''}\) and \(x^2 + y^2 - 10x + 16y = 80\) simultaneously and choosing the coordinate with the greatest modulus. The method is only scored when the distance of the largest coordinate from \(O\) is attempted. Such methods are unlikely to result in an exact value but can score 1 mark for the method. Condone slips
FYI. Solving \(y = -\dfrac{8}{5}x\) and \(x^2 + y^2 - 10x + 16y = 80 \Rightarrow 89x^2 - 890x - 2000 = 0 \Rightarrow P = (11.89,\ -19.02)\)
Hence \(OP = \sqrt{\text{``}11.89\text{''}^2 + \text{``}19.02\text{''}^2}\ (= 22.43)\) scores M1 A0 but \(OP = \sqrt{258 + 26\sqrt{89}}\) is M1 A1