| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Two circles intersection or tangency |
| Difficulty | Moderate -0.3 Part (a) requires routine completion of the square to find centre and radius from general form—standard textbook exercise. Part (b) involves calculating the distance between centres and comparing with sum/difference of radii, which is a straightforward application of the circle intersection condition. While it requires a proof with reasoning, the method is direct and commonly practiced, making this slightly easier than average. |
| 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 |
| \(x^2+y^2+8x-10y=29\); attempts \((x+4)^2+(y-5)^2=\ldots\) | M1 | Complete the square for both variables; may be implied by \((\pm4,\pm5)\) by inspection |
| Correct centre \((-4,5)\) | A1 | |
| Exact radius \(\sqrt{70}\) | A1 | May be scored following \((x\pm4)^2+(y\pm5)^2=70\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts distance (or distance\(^2\)) from \((-4,5)\) to \((5,-8)\) | M1 | Condone sign errors |
| Full method: compares \(\sqrt{9^2+13^2}\) to \((\sqrt{70}+\sqrt{52})\) | dM1 | |
| Distance between centres \(=\sqrt{250}=15.81\); sum of radii \(=\sqrt{70}+\sqrt{52}=15.58\); concludes sum of radii \(<\) distance between centres | A1* | Requires correct calculations, correct reason and minimal conclusion; must show decimal values not just inequality; may use difference between sum of radii and distance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to solve both circle equations simultaneously forming linear equation | M1 | Linear equation is \(18x-26y=66\) if calculations correct |
| Attempts to solve \(18x-26y=66\) simultaneously with either circle equation reaching 3-term quadratic | dM1 | |
| Correct equations; correct "solution" stating no roots and concluding no intersection | A1* | e.g. discriminant \(b^2-4ac=-308256<0\) so no solutions |
# Question 7:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2+y^2+8x-10y=29$; attempts $(x+4)^2+(y-5)^2=\ldots$ | M1 | Complete the square for both variables; may be implied by $(\pm4,\pm5)$ by inspection |
| Correct centre $(-4,5)$ | A1 | |
| Exact radius $\sqrt{70}$ | A1 | May be scored following $(x\pm4)^2+(y\pm5)^2=70$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts distance (or distance$^2$) from $(-4,5)$ to $(5,-8)$ | M1 | Condone sign errors |
| Full method: compares $\sqrt{9^2+13^2}$ to $(\sqrt{70}+\sqrt{52})$ | dM1 | |
| Distance between centres $=\sqrt{250}=15.81$; sum of radii $=\sqrt{70}+\sqrt{52}=15.58$; concludes sum of radii $<$ distance between centres | A1* | Requires correct calculations, correct reason and minimal conclusion; must show decimal values not just inequality; may use difference between sum of radii and distance |
**Alternative:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to solve both circle equations simultaneously forming linear equation | M1 | Linear equation is $18x-26y=66$ if calculations correct |
| Attempts to solve $18x-26y=66$ simultaneously with either circle equation reaching 3-term quadratic | dM1 | |
| Correct equations; correct "solution" stating no roots and concluding no intersection | A1* | e.g. discriminant $b^2-4ac=-308256<0$ so no solutions |
---\begin{enumerate}
\item The circle $C _ { 1 }$ has equation
\end{enumerate}
$$x ^ { 2 } + y ^ { 2 } + 8 x - 10 y = 29$$
(a) (i) Find the coordinates of the centre of $C _ { 1 }$\\
(ii) Find the exact value of the radius of $C _ { 1 }$
In part (b) you must show detailed reasoning.\\
The circle $C _ { 2 }$ has equation
$$( x - 5 ) ^ { 2 } + ( y + 8 ) ^ { 2 } = 52$$
(b) Prove that the circles $C _ { 1 }$ and $C _ { 2 }$ neither touch nor intersect.
\hfill \mbox{\textit{Edexcel P2 2024 Q7 [6]}}