| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Circle touching axes |
| Difficulty | Moderate -0.5 This is a straightforward circle question requiring understanding that a circle touching the y-axis has radius equal to its x-coordinate, then solving a simple equation. Part (a) is direct recall of circle properties, part (b) involves substituting a point and solving a quadratic equation. Slightly easier than average due to minimal problem-solving required. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Writes \(C\) as \((x-a)^2+(y-0)^2=a^2\) | M1A1 | M1: attempts equation with centre \((a,0)\) radius \(a\); accept \((x\pm a)^2+y^2=a^2\). A1: \((x-a)^2+(y-0)^2=a^2\) or equivalent \(x^2+y^2-2ax=0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Subs \((4,-3)\): \((4-a)^2+(-3-0)^2=a^2\) | M1 | Subs \(x=4\), \(y=-3\) into circle equation of form \((x\pm a)^2+(y\pm0)^2=a^2\) |
| \(\Rightarrow 16-8a+a^2+9=a^2 \Rightarrow 25=8a\) | dM1 | Proceeds to linear equation in \(a\); condone numerical slips |
| \(\Rightarrow a=\frac{25}{8}\) | A1 | Accept exact alternatives |
## Question 12:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Writes $C$ as $(x-a)^2+(y-0)^2=a^2$ | M1A1 | M1: attempts equation with centre $(a,0)$ radius $a$; accept $(x\pm a)^2+y^2=a^2$. A1: $(x-a)^2+(y-0)^2=a^2$ or equivalent $x^2+y^2-2ax=0$ |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Subs $(4,-3)$: $(4-a)^2+(-3-0)^2=a^2$ | M1 | Subs $x=4$, $y=-3$ into circle equation of form $(x\pm a)^2+(y\pm0)^2=a^2$ |
| $\Rightarrow 16-8a+a^2+9=a^2 \Rightarrow 25=8a$ | dM1 | Proceeds to linear equation in $a$; condone numerical slips |
| $\Rightarrow a=\frac{25}{8}$ | A1 | Accept exact alternatives |12.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-18_636_887_274_534}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a circle $C$\\
$C$ touches the $y$-axis and has centre at the point ( $a , 0$ ) where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Write down an equation for $C$ in terms of $a$
Given that the point $P ( 4 , - 3 )$ lies on $C$,
\item find the value of $a$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2016 Q12 [5]}}