| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2024 |
| Session | October |
| Marks | 11 |
| Topic | Circles |
| Type | Distance from centre to line |
| Difficulty | Moderate -0.3 This is a straightforward coordinate geometry question testing standard techniques: substituting a line into a circle equation to check for intersection, finding perpendicular lines through a point, and calculating distances. Part (a) is routine substitution and discriminant checking. Part (b) requires completing the square to find the centre, using the perpendicular gradient rule (m₁m₂ = -1), and applying the distance formula. While multi-step, all techniques are standard A-level content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
A line has equation $y = 2x$ and a circle has equation $x^2 + y^2 + 2x - 16y + 56 = 0$.
\begin{enumerate}[label=(\alph*)]
\item Show that the line does not meet the circle. [3]
\item \begin{enumerate}[label=(\roman*)]
\item Find the equation of the line through the centre of the circle that is perpendicular to the line $y = 2x$. [4]
\item Hence find the shortest distance between the line $y = 2x$ and the circle, giving your answer in an exact form. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2024 Q5 [11]}}