| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2024 |
| Session | October |
| Marks | 5 |
| Topic | Circles |
| Type | Two circles intersection or tangency |
| Difficulty | Standard +0.3 This question requires completing the square to find the center and radius of circle C, then using the constraint that two circles of equal radius touch at one point (centers are 2r apart) along a line of gradient 1. While it involves multiple steps, each is a standard technique: completing the square, distance formula, and working with gradient constraints. The problem is straightforward once the setup is understood, making it slightly easier than average. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
A circle, C, has equation $x^2 - 6x + y^2 = 16$.
A second circle, D, has the following properties:
\begin{itemize}
\item The line through the centres of circle C and circle D has gradient 1.
\item Circle D touches circle C at exactly one point.
\item The centre of circle D lies in the first quadrant.
\item Circle D has the same radius as circle C.
\end{itemize}
Find the coordinates of the centre of circle D. [5]
\hfill \mbox{\textit{SPS SPS SM 2024 Q8 [5]}}