| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | October |
| Marks | 11 |
| Topic | Laws of Logarithms |
| Type | Solve exponential equation using logarithms |
| Difficulty | Moderate -0.3 This is a standard coordinate geometry question combining circles and lines. Part (a) requires substitution and showing a discriminant is negative (routine). Part (b) involves finding a circle's centre by completing the square, finding a perpendicular line (gradient = -1/2), and using perpendicular distance formula. All techniques are standard A-level procedures with no novel insight required, 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 |
5. A line has equation $y = 2 x$ and a circle has equation $x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0$.
\begin{enumerate}[label=(\alph*)]
\item Show that the line does not meet the circle.
\item \begin{enumerate}[label=(\roman*)]
\item Find the equation of the line through the centre of the circle that is perpendicular to the line $y = 2 x$.
\item Hence find the shortest distance between the line $y = 2 x$ and the circle, giving your answer in an exact form.\\[0pt]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q5 [11]}}