| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2022 |
| Session | November |
| Marks | 10 |
| Topic | Proof by induction |
| Type | Deduce result from proven formula |
| Difficulty | Moderate -0.8 This is a straightforward coordinate geometry question requiring standard techniques: calculating gradients to verify perpendicularity, finding the circle center as midpoint of diameter (using the angle-in-semicircle theorem), and converting to general form. Despite being labeled 'proof by induction,' it's actually basic circle geometry with routine calculations and no inductive reasoning required. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors |
8. The points $P , Q$ and $R$ have coordinates $( - 5,2 ) , ( - 3,8 )$ and $( 9,4 )$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Show that $\angle P Q R = 90 ^ { \circ }$.
Given that $P , Q$ and $R$ all lie on circle $C$,
\item find the coordinates of the centre of $C$,
\item show that the equation of $C$ can be written in the form
$$x ^ { 2 } + y ^ { 2 } + a x + b y = k$$
where $a , b$ and $k$ are integers to be found.\\[0pt]
\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2022 Q8 [10]}}