| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2025 |
| Session | November |
| Marks | 11 |
| Topic | Circles |
| Type | Two circles intersection or tangency |
| Difficulty | Standard +0.3 This is a standard circles question requiring completion of the square to find centres and radii, checking the distance condition for internal tangency (|c₁c₂| = |r₁ - r₂|), finding the point of contact using section formula, and determining the tangent perpendicular to the line of centres. All techniques are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
The circles $C_1$ and $C_2$ have respective equations
$$x^2 + y^2 - 6x - 2y = 15$$
$$x^2 + y^2 - 18x + 14y = 95.$$
\begin{enumerate}[label=(\alph*)]
\item By considering the coordinates of the centres and the lengths of the radii of $C_1$ and $C_2$, show that $C_1$ and $C_2$ touch internally at some point $P$. [4]
\item Determine the coordinates of $P$. [3]
\item Find the equation of the common tangent to the circles at P. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2025 Q8 [11]}}