| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Two circles intersection or tangency |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on circle equations and intersection points. Part (i) requires writing standard circle equations from given information (likely centers and radii), part (ii) involves solving simultaneous equations to find x-coordinates (typically by subtraction to eliminate one variable), and part (iii) requires substitution to find y-coordinates. While it requires multiple steps, these are all standard C1 techniques with no novel problem-solving or geometric insight needed, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| For A: \((x-1)^2+(y-4)^2=64\) | B1 B1 | LHS, RHS |
| For B: Radius \(=4 \Rightarrow (x-9)^2+(y-4)^2=16\) | B1 B1 |
| Answer | Marks |
|---|---|
| They meet when \(x^2+y^2-2x-8y-47 = x^2+y^2-18x-8y+81\) | M1 A1 |
| \(\Rightarrow -2x-47=-18x+81\) | A1 |
| \(\Rightarrow x=8\) |
| Answer | Marks |
|---|---|
| Substitute in either equation: \(64+y^2-16-8y-47=0\) | M1 A1 |
| \(\Rightarrow y^2-8y+1=0\) | A1 |
| \(\Rightarrow y = \frac{8\pm\sqrt{64-4}}{2} = 4\pm\sqrt{15}\) | M1 A1 |
## Question 10:
**(i)**
For A: $(x-1)^2+(y-4)^2=64$ | B1 B1 | LHS, RHS
For B: Radius $=4 \Rightarrow (x-9)^2+(y-4)^2=16$ | B1 B1 |
**(ii)**
They meet when $x^2+y^2-2x-8y-47 = x^2+y^2-18x-8y+81$ | M1 A1 |
$\Rightarrow -2x-47=-18x+81$ | A1 |
$\Rightarrow x=8$ | |
**(iii)**
Substitute in either equation: $64+y^2-16-8y-47=0$ | M1 A1 |
$\Rightarrow y^2-8y+1=0$ | A1 |
$\Rightarrow y = \frac{8\pm\sqrt{64-4}}{2} = 4\pm\sqrt{15}$ | M1 A1 |
---(i) Write down the equations of the circles A and B .\\
(ii) Find the $x$ coordinates of the points where the two curves intersect.\\
(iii) Find the $y$ coordinates of these points, giving your answers in surd form.
\hfill \mbox{\textit{OCR MEI C1 Q10 [12]}}