| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 11 |
| Topic | Circles |
| Type | Range of parameter for intersection |
| Difficulty | Standard +0.3 This is a standard discriminant problem for circle-line intersection. Part (a) requires substituting the line into the circle equation and applying b²-4ac>0, which is routine A-level technique. Part (a)(ii) involves solving a quadratic inequality. Parts (b) are straightforward sketching and observation. Slightly easier than average due to being a well-practiced question type with clear methodology. |
| 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 |
6.
A curve has equation $x ^ { 2 } + y ^ { 2 } + 12 x = 64$\\
A line has equation $y = m x + 10$
\begin{enumerate}[label=(\alph*)]
\item (i) In the case that the line intersects the curve at two distinct points, show that
$$( 20 m + 12 ) ^ { 2 } - 144 \left( m ^ { 2 } + 1 \right) > 0$$
(a) (ii) Hence find the possible values of $m$.
\item (i) On the same diagram, sketch the curve and the line in the case when $m = 0$\\
(b) (ii) State the relationship between the curve and the line.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q6 [11]}}