| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2004 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Circles |
| Type | Circle touching axes |
| Difficulty | Challenging +1.8 This AEA question requires finding a circle's radius using perpendicular distance from center to tangent line, proving a non-trivial trigonometric identity involving inverse tangent functions (requiring angle addition formulas and geometric insight), and finding another tangent line equation. The trigonometric identity proof in part (a)(ii) is the challenging element, requiring students to recognize geometric relationships or manipulate arctan expressions algebraically—well above routine A-level but not requiring exceptionally deep insight. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03f Circle properties: angles, chords, tangents1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 |
\includegraphics{figure_1}
The circle, with centre $C$ and radius $r$, touches the $y$-axis at $(0, 4)$ and also touches the line with equation $4y - 3x = 0$, as shown in Fig. 1.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the value of $r$.
\item Show that $\arctan \left(\frac{4}{3}\right) + 2 \arctan \left(\frac{1}{2}\right) = \frac{1}{2} \pi$. [8]
\end{enumerate}
\end{enumerate}
The line with equation $4x + 3y = q$, $q > 12$, is a tangent to the circle.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the value of $q$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2004 Q4 [12]}}