| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Geometric properties with circles |
| Difficulty | Standard +0.3 This is a straightforward application of coordinate transformations to circles. Part (a) requires understanding that a horizontal stretch transforms a circle into an ellipse and finding new intersection points (routine calculation). Part (b)(i) involves substituting the transformation into the circle equation to get the ellipse equation (standard technique). Part (b)(ii) requires finding the tangent using either implicit differentiation or the standard ellipse tangent formula. All steps are mechanical applications of well-practiced techniques with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| 6(a) Sketch of ellipse | M1 | |
| Correct relationship to circle | A1 | |
| Coords \(\left(\pm 2\sqrt{2}, 0\right), \left(0, \pm\sqrt{2}\right)\) | B2, 1 | 4 marks |
| 6(b)(i) Replacing \(x\) by \(\frac{x}{2}\) | M1 | |
| \(E\) is \(\left(\frac{x}{2}\right)^2 + y^2 = 2\) | A1 | 2 marks |
| 6(b)(ii) Tangent is \(\frac{x}{2} + y = 2\) | M1A1 | 2 marks |
**6(a)** Sketch of ellipse | M1 | | centred at origin
Correct relationship to circle | A1 | |
Coords $\left(\pm 2\sqrt{2}, 0\right), \left(0, \pm\sqrt{2}\right)$ | B2, 1 | 4 marks | Accept $\sqrt{8}$ for $2\sqrt{2}$; B1 for any 2 of $x = \pm 2\sqrt{2}, y = \pm \sqrt{2}$; allow B1 if all correct except for use of decimals (at least one DP)
**6(b)(i)** Replacing $x$ by $\frac{x}{2}$ | M1 | | or by $2x$
$E$ is $\left(\frac{x}{2}\right)^2 + y^2 = 2$ | A1 | 2 marks | OE
**6(b)(ii)** Tangent is $\frac{x}{2} + y = 2$ | M1A1 | 2 marks | M1 for complete valid method
**Total: 8 marks**
---6 The diagram shows a circle $C$ and a line $L$, which is the tangent to $C$ at the point $( 1,1 )$. The equations of $C$ and $L$ are
$$x ^ { 2 } + y ^ { 2 } = 2 \text { and } x + y = 2$$
respectively.\\
\includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_760_1301_552_395}
The circle $C$ is now transformed by a stretch with scale factor 2 parallel to the $x$-axis. The image of $C$ under this stretch is an ellipse $E$.
\begin{enumerate}[label=(\alph*)]
\item On the diagram below, sketch the ellipse $E$, indicating the coordinates of the points where it intersects the coordinate axes.
\item Find equations of:
\begin{enumerate}[label=(\roman*)]
\item the ellipse $E$;
\item the tangent to $E$ at the point $( 2,1 )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_743_1301_1921_420}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2011 Q6 [8]}}