| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Line-circle intersection points |
| Difficulty | Moderate -0.8 This is a routine C1 circle question testing standard techniques: completing the square (straightforward with even coefficients), reading off centre/radius, substituting a linear equation into the circle equation, and solving a simple quadratic that factors easily to (x-2)(x-3)=0. All steps are algorithmic with no problem-solving or insight required, making it easier than average but not trivial since it requires careful algebraic manipulation across multiple parts. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
3 A circle has equation $x ^ { 2 } + y ^ { 2 } - 12 x - 6 y + 20 = 0$.
\begin{enumerate}[label=(\alph*)]
\item By completing the square, express the equation in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
\item Write down:
\begin{enumerate}[label=(\roman*)]
\item the coordinates of the centre of the circle;
\item the radius of the circle.
\end{enumerate}\item The line with equation $y = x + 4$ intersects the circle at the points $P$ and $Q$.
\begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinates of $P$ and $Q$ satisfy the equation
$$x ^ { 2 } - 5 x + 6 = 0$$
\item Find the coordinates of $P$ and $Q$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2005 Q3 [11]}}