| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Hyperbola focus-directrix properties |
| Difficulty | Standard +0.3 This is a straightforward application of standard ellipse formulas requiring identification of a² and b², then using c² = a² - b² for foci and x = ±a²/c for directrices. While it's Further Maths content, it involves direct formula recall with minimal problem-solving, making it slightly easier than an average A-level question overall. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Uses \(b^2 = a^2(1-e^2)\) to find a value of \(e\); look for \(20 = 36(1-e^2)\) | M1 | Uses \(b^2 = a^2(1-e^2)\) to obtain a value of \(e\) (allow if \(-\frac{2}{3}\) also given) |
| \(e = \frac{2}{3} \Rightarrow\) foci are \((\pm 6 \times e, 0)\) | dM1 | Uses \(a=6\) and their value of \(e\) with \(0 < e < 1\), to find at least one focus using \(((\pm)ae, 0)\) |
| Foci are \((\pm 4, 0)\) | A1 | Correct foci — both required, including \(y\) coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Sets up equation \(2\sqrt{p^2 + b^2} = 2a\) where \(p\) is the \(x\)-coordinate of the foci; \(2\sqrt{p^2 + 20} = 12\) | M1 | Sets up equation using total distance from foci to point on ellipse \(= 2a\) |
| Solves to find value of \(p\) | dM1 | Solves to find the \(x\) coordinate of the foci |
| Foci are \((\pm 4, 0)\) | A1 | Correct foci — both required, including \(y\) coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Directrices are \(x = (\pm)\dfrac{6}{\text{their } e}\) | M1 | Uses \(x = (\pm)\dfrac{a}{e}\) with \(a=6\) and their \(e\) to attempt directrices |
| \(x = \pm 9\) only | A1 | Correct directrices, both required and no other lines |
# Question 1:
## Part (a)
| Working/Answer | Mark | Guidance |
|---|---|---|
| Uses $b^2 = a^2(1-e^2)$ to find a value of $e$; look for $20 = 36(1-e^2)$ | M1 | Uses $b^2 = a^2(1-e^2)$ to obtain a value of $e$ (allow if $-\frac{2}{3}$ also given) |
| $e = \frac{2}{3} \Rightarrow$ foci are $(\pm 6 \times e, 0)$ | dM1 | Uses $a=6$ and their value of $e$ with $0 < e < 1$, to find at least one focus using $((\pm)ae, 0)$ |
| Foci are $(\pm 4, 0)$ | A1 | Correct foci — both required, including $y$ coordinates |
**Alternative:**
| Working/Answer | Mark | Guidance |
|---|---|---|
| Sets up equation $2\sqrt{p^2 + b^2} = 2a$ where $p$ is the $x$-coordinate of the foci; $2\sqrt{p^2 + 20} = 12$ | M1 | Sets up equation using total distance from foci to point on ellipse $= 2a$ |
| Solves to find value of $p$ | dM1 | Solves to find the $x$ coordinate of the foci |
| Foci are $(\pm 4, 0)$ | A1 | Correct foci — both required, including $y$ coordinates |
## Part (b)
| Working/Answer | Mark | Guidance |
|---|---|---|
| Directrices are $x = (\pm)\dfrac{6}{\text{their } e}$ | M1 | Uses $x = (\pm)\dfrac{a}{e}$ with $a=6$ and their $e$ to attempt directrices |
| $x = \pm 9$ only | A1 | Correct directrices, both required and no other lines |
---\begin{enumerate}
\item The ellipse $E$ has equation
\end{enumerate}
$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$$
Find\\
(a) the coordinates of the foci of $E$,\\
(b) the equations of the directrices of $E$.
\hfill \mbox{\textit{Edexcel FP1 2021 Q1 [5]}}