| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Point position relative to circle |
| Difficulty | Moderate -0.8 This is a straightforward question requiring completing the square to find centre and radius (standard technique), then using the distance formula. Part (b) only requires recognizing that the furthest point lies on the line through O and the centre, making it centre distance plus radius. All steps are routine with no problem-solving insight needed. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) \(x^2+y^2-10x+16y=80 \Rightarrow (x-5)^2+(y+8)^2 = \ldots\) | M1 | Attempts to complete the square on both \(x\) and \(y\) terms. Accept \((x\pm5)^2+(y\pm8)^2=\ldots\) or imply for centre \((\pm5,\pm8)\). Condone \((x\pm5)^2\ldots(y\pm8)^2=\ldots\) where first \(\ldots\) could be \(,\) or even \(-\) |
| Centre \((5,-8)\) | A1 | Accept without brackets; may be written \(x=5, y=-8\) |
| (ii) Radius \(13\) | A1 | M mark must have been awarded. Do not allow \(\sqrt{169}\) or \(\pm13\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts \(\sqrt{"5"^2 + "8"^2} + "13"\) | M1 | Award when given as 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 \(r\) |
| \(13 + \sqrt{89}\) | A1ft | Follow through on their \((5,-8)\) and their \(13\) leading to exact answer. ISW if they write \(13+\sqrt{89}=22.4\) |
## Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| (i) $x^2+y^2-10x+16y=80 \Rightarrow (x-5)^2+(y+8)^2 = \ldots$ | M1 | Attempts to complete the square on **both** $x$ and $y$ terms. Accept $(x\pm5)^2+(y\pm8)^2=\ldots$ or imply for centre $(\pm5,\pm8)$. Condone $(x\pm5)^2\ldots(y\pm8)^2=\ldots$ where first $\ldots$ could be $,$ or even $-$ |
| Centre $(5,-8)$ | A1 | Accept without brackets; may be written $x=5, y=-8$ |
| (ii) Radius $13$ | A1 | M mark must have been awarded. Do not allow $\sqrt{169}$ or $\pm13$ |
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## Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts $\sqrt{"5"^2 + "8"^2} + "13"$ | M1 | Award when given as 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 $r$ |
| $13 + \sqrt{89}$ | A1ft | Follow through on their $(5,-8)$ and their $13$ leading to exact answer. ISW if they write $13+\sqrt{89}=22.4$ |
---\begin{enumerate}
\item A circle has equation
\end{enumerate}
$$x ^ { 2 } + y ^ { 2 } - 10 x + 16 y = 80$$
(a) Find\\
(i) the coordinates of the centre of the circle,\\
(ii) the radius of the circle.
Given that $P$ is the point on the circle that is furthest away from the origin $O$,\\
(b) find the exact length $O P$
\hfill \mbox{\textit{Edexcel Paper 1 2022 Q3 [5]}}