| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2022 |
| Session | June |
| Marks | 24 |
| Paper | Download PDF ↗ |
| Topic | Circles |
| Type | Find equations of tangent lines with given gradient or from external point using discriminant |
| Difficulty | Challenging +1.8 This AEA question requires systematic application of the discriminant condition for tangency, followed by solving a system involving two circles. Parts (a)-(b) are guided algebraic derivations. Parts (c)-(d) demand geometric insight about normals/tangents to two circles and solving the resulting system of equations with the given Pythagorean triple hint. While lengthy and requiring careful algebra, the problem-solving path is relatively structured for an AEA question, making it challenging but not at the extreme end. |
| Spec | 1.03b 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 |
7.A circle $C$ has centre $X ( a , b )$ and radius $r$ .\\
A line $l$ has equation $y = m x + c$
\begin{enumerate}[label=(\alph*)]
\item Show that the $x$ coordinates of the points where $C$ and $l$ intersect satisfy
$$\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 2 ( a - m ( c - b ) ) x + a ^ { 2 } + ( c - b ) ^ { 2 } - r ^ { 2 } = 0$$
Given that $l$ is a tangent to $C$ ,
\item show that
$$c = b - m a \pm r \sqrt { m ^ { 2 } + 1 }$$
The circle $C _ { 1 }$ has equation
$$x ^ { 2 } + y ^ { 2 } - 16 = 0$$
and the circle $C _ { 2 }$ has equation
$$x ^ { 2 } + y ^ { 2 } - 20 x - 10 y + 89 = 0$$
\item Find the equations of any lines that are normal to both $C _ { 1 }$ and $C _ { 2 }$ ,justifying your answer.
\item Find the equations of all lines that are a tangent to both $C _ { 1 }$ and $C _ { 2 }$\\
[You may find the following Pythagorean triple helpful in this part: $7 ^ { 2 } + 24 ^ { 2 } = 25 ^ { 2 }$ ]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2022 Q7 [24]}}