| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Circle touching axes |
| Difficulty | Moderate -0.8 This is a straightforward multi-part circle question testing standard techniques: finding radius from tangency condition, writing circle equation, finding gradient, using perpendicular gradient for tangent, and applying perpendicular distance to chord formula. All parts are routine C1 exercises with no problem-solving insight required, making it easier than average. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03f Circle properties: angles, chords, tangents1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| \((x - 8)^2 + (y - 13)^2\) | B1 | |
| \(= 13^2\) | B1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| grad \(PC = \frac{12}{5}\) | B1 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| grad of tangent \(= \frac{-1}{\text{grad } PC} = -\frac{5}{12}\) | B1 | |
| tangent has equation \(y - 1 = -\frac{5}{12}(x - 3)\) | M1, A1 | |
| \(5x + 12y = 27\) OE | A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| half chord \(= 5\) | B1 | |
| \(d^2 = (\text{their } r)^2 - 5^2\) (provided \(r > 5\)) | M1 | |
| Distance \(= 12\) | A1 | 3 |
**7(a)**
| $(x - 8)^2 + (y - 13)^2$ | B1 | | **Exactly this with $+$ and squares** |
| $= 13^2$ | B1 | 2 | Condone 169 |
**7(b)(i)**
| grad $PC = \frac{12}{5}$ | B1 | 1 | Must simplify $\frac{-12}{-5}$ |
**7(b)(ii)**
| grad of tangent $= \frac{-1}{\text{grad } PC} = -\frac{5}{12}$ | B1 | | Condone $-\frac{1}{2.4}$ etc |
| tangent has equation $y - 1 = -\frac{5}{12}(x - 3)$ | M1, A1 | | ft gradient but M0 if using grad $PC$ Correct – but not in required final form |
| $5x + 12y = 27$ OE | A1 | 4 | **MUST have integer coefficients** |
**7(b)(iii)**
| half chord $= 5$ | B1 | | Seen or stated |
| $d^2 = (\text{their } r)^2 - 5^2$ (provided $r > 5$) | M1 | | Pythagoras used correctly $d^2 = 13^2 - 5^2$ |
| Distance $= 12$ | A1 | 3 | CSO |
---7 The circle $S$ has centre $C ( 8,13 )$ and touches the $x$-axis, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-4_444_755_356_641}
\begin{enumerate}[label=(\alph*)]
\item Write down an equation for $S$, giving your answer in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
\item The point $P$ with coordinates $( 3,1 )$ lies on the circle.
\begin{enumerate}[label=(\roman*)]
\item Find the gradient of the straight line passing through $P$ and $C$.
\item Hence find an equation of the tangent to the circle $S$ at the point $P$, giving your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers.
\item The point $Q$ also lies on the circle $S$, and the length of $P Q$ is 10 . Calculate the shortest distance from $C$ to the chord $P Q$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2008 Q7 [10]}}