| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2025 |
| Session | October |
| Marks | 9 |
| Topic | Circles |
| Type | Find equations of tangent lines with given gradient or from external point using discriminant |
| Difficulty | Standard +0.3 This is a standard circle question requiring completion of the square to find centre and radius, then applying geometric conditions for intersection and tangency. Part (a) is routine manipulation, part (b) uses the radius to find the range of y-values, and part (c) requires solving a discriminant condition for tangency. All techniques are standard A-level fare with no novel insight required, making it slightly easier than average. |
| Spec | 1.02h Express solutions: using 'and', 'or', set and interval notation1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
The circle $C$ has equation
$$x^2 + y^2 + 10x - 4y + 1 = 0$$
\begin{enumerate}[label=(\alph*)]
\item Find
\begin{enumerate}[label=(\roman*)]
\item the coordinates of the centre of $C$
\item the exact radius of $C$ [2]
\end{enumerate}
The line with equation $y = k$, where $k$ is a constant, cuts $C$ at two distinct points.
\item Find the range of values for $k$, giving your answer in set notation. [2]
\item The line with equation $y = mx + 4$ is a tangent to $C$. Find possible exact values of $m$. [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2025 Q13 [9]}}