| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Topic | Circles |
| Type | Line-circle intersection points |
| Difficulty | Easy -1.2 This is a straightforward coordinate geometry question testing standard circle equation manipulation and intersection with lines. Part (a) is direct substitution, part (b) requires setting y=0 and solving a quadratic, and part (c) involves substituting a linear equation into the circle equation—all routine A-level techniques with no novel problem-solving required. The 'show that' format provides the target equation, making it easier than open-ended questions. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
A circle has centre $C(3, -8)$ and radius $10$.
\begin{enumerate}[label=(\alph*)]
\item Express the equation of the circle in the form
$$(x - a)^2 + (y - b)^2 = k$$ [2 marks]
\item Find the $x$-coordinates of the points where the circle crosses the $x$-axis. [3 marks]
\item The line with equation $y = 2x + 1$ intersects the circle.
\begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinates of the points of intersection satisfy the equation
$$x^2 + 6x - 2 = 0$$ [3 marks]
\item Hence show that the $x$-coordinates of the points of intersection are of the form $m \pm \sqrt{n}$, where $m$ and $n$ are integers. [2 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2022 Q3 [10]}}