| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Circle from diameter endpoints |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard circle properties: midpoint formula for the centre, distance formula for radius, and the geometric property that angles in a semicircle are 90°. All parts follow routine procedures with no problem-solving insight required, making it easier than average but not trivial due to the algebraic manipulation needed in part (b). |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(= (-\frac{2+4}{2}, \frac{6-1}{2}) = (1, \frac{5}{2})\) | M1 A1 | |
| (b) radius \(= \text{dist. } (-2, 6)\) to \((1, \frac{5}{2}) = \sqrt{9 + \frac{49}{4}} = \sqrt{\frac{85}{4}}\) | M1 A1 | |
| \(\therefore (x-1)^2 + (y - \frac{5}{2})^2 = (\sqrt{\frac{85}{4}})^2\) | M1 A1 | |
| \(x^2 - 2x + 1 + y^2 - 5y + \frac{25}{4} = \frac{85}{4}\) | ||
| \(x^2 + y^2 - 2x - 5y - 14 = 0\) | A1 | |
| (c) \((2, 7)\): LHS \(= 4 + 49 - 4 - 35 - 14 = 0 \therefore R\) lies on circle | B1 | |
| \(\angle PRQ = 90°\) | B1 | (9 marks) |
(a) $= (-\frac{2+4}{2}, \frac{6-1}{2}) = (1, \frac{5}{2})$ | M1 A1 |
(b) radius $= \text{dist. } (-2, 6)$ to $(1, \frac{5}{2}) = \sqrt{9 + \frac{49}{4}} = \sqrt{\frac{85}{4}}$ | M1 A1 |
$\therefore (x-1)^2 + (y - \frac{5}{2})^2 = (\sqrt{\frac{85}{4}})^2$ | M1 A1 |
$x^2 - 2x + 1 + y^2 - 5y + \frac{25}{4} = \frac{85}{4}$ | |
$x^2 + y^2 - 2x - 5y - 14 = 0$ | A1 |
(c) $(2, 7)$: LHS $= 4 + 49 - 4 - 35 - 14 = 0 \therefore R$ lies on circle | B1 |
$\angle PRQ = 90°$ | B1 | (9 marks)7. The points $P$ and $Q$ have coordinates $( - 2,6 )$ and $( 4 , - 1 )$ respectively.
Given that $P Q$ is a diameter of circle $C$,
\begin{enumerate}[label=(\alph*)]
\item find the coordinates of the centre of $C$,
\item show that $C$ has the equation
$$x ^ { 2 } + y ^ { 2 } - 2 x - 5 y - 14 = 0 .$$
The point $R$ has coordinates (2, 7).
\item Show that $R$ lies on $C$ and hence, state the size of $\angle P R Q$ in degrees.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q7 [9]}}