| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Circle from diameter endpoints |
| Difficulty | Easy -1.2 This is a straightforward C2 circle question requiring only standard formulas: distance formula for diameter length, midpoint formula for center, and circle equation from center and radius. All three parts are direct applications of memorized techniques with no problem-solving or geometric insight required. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((AB)^2 = (4-3)^2 + (5)^2\) \([=26]\) | M1 | |
| \(AB = \sqrt{26}\) | A1 | (2 marks) |
| (b) \(p = \left(\frac{4+3}{2}, \frac{5}{2}\right)\) | M1 | |
| \(= \left(\frac{7}{2}, \frac{5}{2}\right)\) | A1 | (2 marks) |
| (c) \((x-x_p)^2 + (y-y_p)^2 = \left(\frac{AB}{2}\right)^2\) | LHS M1; RHS M1 | |
| \((x-3.5)^2 + (y-2.5)^2 = 6.5\) | oe A1 c.a.o | (3 marks) |
(a) $(AB)^2 = (4-3)^2 + (5)^2$ $[=26]$ | M1
$AB = \sqrt{26}$ | A1 | (2 marks)
(b) $p = \left(\frac{4+3}{2}, \frac{5}{2}\right)$ | M1
$= \left(\frac{7}{2}, \frac{5}{2}\right)$ | A1 | (2 marks)
(c) $(x-x_p)^2 + (y-y_p)^2 = \left(\frac{AB}{2}\right)^2$ | LHS M1; RHS M1
$(x-3.5)^2 + (y-2.5)^2 = 6.5$ | oe A1 c.a.o | (3 marks)
**Guidance:**
- (a) M1 for an expression for $AB$ or $AB^2$. N.B. $(x_1+x_2)^2 + \ldots$ is M0
- (b) M1 for a full method for $x_p$
- (c) 1st M1 for using their $x_p$ and $y_p$ in LHS; 2nd M1 for using their $AB$ in RHS
**Note:** $x^2 + y^2 - 7x - 5y + 12 = 0$ scores 3/3 for part (c).
Condone use of calculator approximations that lead to correct answer given.
**Total: 7 marks**
---3.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{84b2d36b-c112-4d35-84a1-bc2b707f162d-04_675_792_287_568}
\end{center}
\end{figure}
In Figure $1 , A ( 4,0 )$ and $B ( 3,5 )$ are the end points of a diameter of the circle $C$.
Find
\begin{enumerate}[label=(\alph*)]
\item the exact length of $A B$,
\item the coordinates of the midpoint $P$ of $A B$,
\item an equation for the circle $C$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2006 Q3 [7]}}