A2 June 2023 Paper 2 Q5
5 Josh and Zoe are solving the following mathematics problem:
The curve \(C_1\) has equation \[\frac{x^2}{16} - \frac{y^2}{9} = 1\]The matrix \(\mathbf{M} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) maps \(C_1\) onto \(C_2\) Find the equations of the asymptotes of \(C_2\) |
Josh says that to solve this problem you must first carry out the transformation on \(C_1\) to find \(C_2\), and then find the asymptotes of \(C_2\)
Zoe says that you will get the same answer if you first find the asymptotes of \(C_1\), and then carry out the transformation on these asymptotes to obtain the asymptotes of \(C_2\)
Show that Zoe is correct. [5 marks]
| Scheme | Marks | AO |
|---|---|---|
| States the correct asymptotes of \(C_1\) | B1 | 1.1b |
| States the correct equation of \(C_2\) | B1 | 3.1a |
| States the correct asymptotes of \(C_2\) | B1 | 1.1b |
| Obtains the asymptotes of \(C_2\) by both methods. | M1 | 3.1a |
| Shows that both methods lead to the same answer and concludes that Zoe is correct. | R1 | 2.3 |
| (5 marks) |
Typical solution
Josh’s method
Reflection in \(y = x\)
\[C_2 \text{ is } \frac{y^2}{16} - \frac{x^2}{9} = 1\]The asymptotes of \(C_2\) are \(y = \pm\dfrac{4}{3}x\)
Zoe’s method
The asymptotes of \(C_1\) are \(y = \pm\dfrac{3}{4}x\)
The transformation is a reflection in \(y = x\)
The asymptotes of \(C_2\) are \(y = \pm\dfrac{4}{3}x\)
Both answers are the same, so Zoe is correct.