| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2006 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Circles |
| Type | Tangent from external point - intersection or geometric properties |
| Difficulty | Challenging +1.2 This is an AEA question involving tangents to circles from external points, requiring perpendicular distance formula, solving quadratics, and coordinate geometry. Part (a) is standard algebraic manipulation, part (b) requires finding tangent points using perpendicularity, and part (c) exploits symmetry with the first circle. While multi-step and requiring careful work, the techniques are all standard A-level methods with no novel insight needed, making it moderately above average difficulty. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.07m Tangents and normals: gradient and equations |
4.The line with equation $y = m x$ is a tangent to the circle $C _ { 1 }$ with equation
$$( x + 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 13$$
\begin{enumerate}[label=(\alph*)]
\item Show that $m$ satisfies the equation
$$3 m ^ { 2 } + 56 m + 36 = 0$$
The tangents from the origin $O$ to $C _ { 1 }$ touch $C _ { 1 }$ at the points $A$ and $B$ .
\item Find the coordinates of the points $A$ and $B$ .\\
(8)\\
Another circle $C _ { 2 }$ has equation $x ^ { 2 } + y ^ { 2 } = 13$ .The tangents from the point $( 4 , - 7 )$ to $C _ { 2 }$ touch it at the points $P$ and $Q$ .
\item Find the coordinates of either the point $P$ or the point $Q$ .\\
(2)
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2006 Q4 [14]}}