| 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 standard C2 circle question requiring the circle equation formula, substitution to find intersection points via a quadratic, and distance formula application. All techniques are routine for this level, though the multi-part structure and algebraic manipulation (solving the resulting quadratic and computing distance) place it slightly below average difficulty rather than being trivial. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((x + 1)^2 + (y - 6)^2 = (2\sqrt{5})^2\) | M1 | |
| \((x + 1)^2 + (y - 6)^2 = 20\) | A1 | |
| (b) Sub. \(y = 3x - 1\) into eqn of \(C\): | M1 | |
| \((x + 1)^2 + [(3x - 1) - 6]^2 = 20\) | A1 | |
| \((x + 1)^2 + (3x - 7)^2 = 20\) | A1 | |
| \(x^2 - 4x + 3 = 0\) | M1 | |
| \((x - 1)(x - 3) = 0\) | M1 | |
| \(x = 1, 3\) | A1 | |
| (c) \(x = 1 \Rightarrow y = 2 \quad \therefore (1,2)\) | B1 | |
| \(x = 3 \Rightarrow y = 8 \quad \therefore (3,8)\) | B1 | |
| \(AB = \sqrt{(3-1)^2 + (8-2)^2} = \sqrt{4 + 36} = \sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}\) | M1 A1 | (9 marks) |
**(a)** $(x + 1)^2 + (y - 6)^2 = (2\sqrt{5})^2$ | M1 |
$(x + 1)^2 + (y - 6)^2 = 20$ | A1 |
**(b)** Sub. $y = 3x - 1$ into eqn of $C$: | M1 |
$(x + 1)^2 + [(3x - 1) - 6]^2 = 20$ | A1 |
$(x + 1)^2 + (3x - 7)^2 = 20$ | A1 |
$x^2 - 4x + 3 = 0$ | M1 |
$(x - 1)(x - 3) = 0$ | M1 |
$x = 1, 3$ | A1 |
**(c)** $x = 1 \Rightarrow y = 2 \quad \therefore (1,2)$ | B1 |
$x = 3 \Rightarrow y = 8 \quad \therefore (3,8)$ | B1 |
$AB = \sqrt{(3-1)^2 + (8-2)^2} = \sqrt{4 + 36} = \sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}$ | M1 A1 | **(9 marks)**\begin{enumerate}
\item The circle $C$ has centre $( - 1,6 )$ and radius $2 \sqrt { 5 }$.\\
(a) Find an equation for $C$.
\end{enumerate}
The line $y = 3 x - 1$ intersects $C$ at the points $A$ and $B$.\\
(b) Find the $x$-coordinates of $A$ and $B$.\\
(c) Show that $A B = 2 \sqrt { 10 }$.\\
\hfill \mbox{\textit{Edexcel C2 Q5 [9]}}