| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Line-circle intersection points |
| Difficulty | Moderate -0.3 This is a straightforward circle-line intersection problem requiring standard techniques: writing circle equation from center/radius, substituting line equation into circle equation to solve a quadratic, then using distance formula. While it involves multiple steps (9 marks total) and some algebraic manipulation with surds, all techniques are routine C2 content with no conceptual challenges or novel insights required. Slightly easier than average due to the mechanical nature of the procedures. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| \((x - 3)^2 + (y - 4)^2 = 18\) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Use \(y = x + 3\) to obtain \((x - 3)^2 + (x - 1)^2 = 18\) | M1 | |
| And thus \(2x^2 - 8x = 8\) | A1 | |
| Solve quadratic, to obtain \(x = 2 \pm \sqrt{8}\), \(y = 5 \pm \sqrt{8}\) | M1, A1ft, A1ft | (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Distance \(= \sqrt{(2\sqrt{8})^2 + (2\sqrt{8})^2} = 8\) | M1 A1 cso | (2 marks) |
**Part (a)**
$(x - 3)^2 + (y - 4)^2 = 18$ | M1 A1 | (2 marks)
**Part (b)**
Use $y = x + 3$ to obtain $(x - 3)^2 + (x - 1)^2 = 18$ | M1 |
And thus $2x^2 - 8x = 8$ | A1 |
Solve quadratic, to obtain $x = 2 \pm \sqrt{8}$, $y = 5 \pm \sqrt{8}$ | M1, A1ft, A1ft | (5 marks)
**Part (c)**
Distance $= \sqrt{(2\sqrt{8})^2 + (2\sqrt{8})^2} = 8$ | M1 A1 cso | (2 marks)
**Total for Question 5: 9 marks**
---A circle $C$ has centre $(3, 4)$ and radius $3\sqrt{2}$. A straight line $l$ has equation $y = x + 3$.
\begin{enumerate}[label=(\alph*)]
\item Write down an equation of the circle $C$. [2]
\item Calculate the exact coordinates of the two points where the line $l$ intersects $C$, giving your answers in surds. [5]
\item Find the distance between these two points. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q5 [9]}}