| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 7 |
| Topic | Circles |
| Type | Tangent equation at a known point on circle |
| Difficulty | Moderate -0.8 This is a straightforward circle question requiring completion of the square to find centre and radius (standard technique), then finding a tangent using the perpendicular radius property. Both parts are routine textbook exercises with no problem-solving insight needed, making it easier than average but not trivial due to the algebraic manipulation required. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.03f Circle properties: angles, chords, tangents |
$$x^2 + y^2 - 2x - 2y = 8$$
The circle with the above equation has radius $r$ and has its centre at the point $C$.
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $r$ and the coordinates of $C$. [3]
\end{enumerate}
The point $P(4,2)$ lies on the circle.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that an equation of the tangent to the circle at $P$ is [4]
$$y = 14 - 3x.$$
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q3 [7]}}