| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | October |
| Marks | 10 |
| Topic | Circles |
| Type | Find centre and radius from equation |
| Difficulty | Moderate -0.8 This is a routine multi-part circle geometry question requiring completing the square, solving simultaneous equations by substitution, and using the midpoint property of diameters. All techniques are standard A-level procedures with no novel problem-solving required, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
The equation of a circle is $x^2 + y^2 + 6x - 2y - 10 = 0$.
\begin{enumerate}[label=(\roman*)]
\item Find the centre and radius of the circle. [3]
\item Find the coordinates of any points where the line $y = 2x - 3$ meets the circle $x^2 + y^2 + 6x - 2y - 10 = 0$. [4]
\item State what can be deduced from the answer to part ii. about the line $y = 2x - 3$ and the circle $x^2 + y^2 + 6x - 2y - 10 = 0$. [1]
\item The point $A(-1,5)$ lies on the circumference of the circle $x^2 + y^2 + 6x - 2y - 10 = 0$. Given that $AB$ is a diameter of the circle, find the coordinates of $B$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q8 [10]}}