| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| 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 hyperbola formulas relating a, b, and the eccentricity. Students need to recall that c² = a² + b² for a hyperbola and use c = 13, b² = 25 to find a, then apply the directrix formula x = ±a/e. It requires recall of formulas and basic algebraic manipulation but no problem-solving insight or multi-step reasoning. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(ae = 13\) and \(a^2(e^2 - 1) = 25\) | B1 | Sight of both (can be implied by work); allow \(\pm ae = \pm 13\) or \(ae = 13\) or \(ae = \pm 13\) |
| Solves to obtain \(a^2 = \ldots\) or \(a = \ldots\) | M1 | Eliminates \(e\) to reach \(a^2 = \ldots\) or \(a = \ldots\) |
| \(a = 12\) | A1 | Cao (not \(\pm 12\)) unless \(-12\) is rejected |
| \(e = 13/\text{"12"}\) | M1 | Uses their \(a\) to find \(e\) or finds \(e\) by eliminating \(a\) (ignore \(\pm\) here); can be implied by correct answer |
| \(x = (\pm)\frac{a}{e} = \pm\frac{144}{13}\) | M1, A1 | M1: \((x=)(\pm)\frac{a}{e}\); \(\pm\) not needed, look for use of \(\frac{a}{e}\) with numerical \(a\) and \(e\). A1: \(x = \pm\frac{144}{13}\) oe, must be an equation (do not allow \(x = \pm\frac{12}{13/12}\)) |
## Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $ae = 13$ **and** $a^2(e^2 - 1) = 25$ | B1 | Sight of **both** (can be implied by work); allow $\pm ae = \pm 13$ or $ae = 13$ or $ae = \pm 13$ |
| Solves to obtain $a^2 = \ldots$ or $a = \ldots$ | M1 | Eliminates $e$ to reach $a^2 = \ldots$ or $a = \ldots$ |
| $a = 12$ | A1 | Cao (not $\pm 12$) unless $-12$ is rejected |
| $e = 13/\text{"12"}$ | M1 | Uses their $a$ to find $e$ or finds $e$ by eliminating $a$ (ignore $\pm$ here); can be implied by correct answer |
| $x = (\pm)\frac{a}{e} = \pm\frac{144}{13}$ | M1, A1 | M1: $(x=)(\pm)\frac{a}{e}$; $\pm$ not needed, look for use of $\frac{a}{e}$ with numerical $a$ and $e$. A1: $x = \pm\frac{144}{13}$ oe, must be an equation (do not allow $x = \pm\frac{12}{13/12}$) |
**Total: 6**
Note: If eccentricity equation for ellipse ($b^2 = a^2(1-e^2)$) is used, allow the M's.
---\begin{enumerate}
\item A hyperbola $H$ has equation
\end{enumerate}
$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1 , \quad \text { where } a \text { is a positive constant. }$$
The foci of $H$ are at the points with coordinates $( 13,0 )$ and $( - 13,0 )$.\\
Find\\
(a) the value of the constant $a$,\\
(b) the equations of the directrices of $H$.\\
\hfill \mbox{\textit{Edexcel FP3 2013 Q1 [6]}}