| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Hyperbola focus-directrix properties |
| Difficulty | Challenging +1.2 This is a Further Maths conic sections question requiring knowledge of hyperbola properties (eccentricity formula e² = 1 + b²/a²) and substitution of a point to create simultaneous equations. While it involves FM content and multiple steps, the solution path is straightforward: use the eccentricity to relate a and b, substitute the point coordinates, solve the system. No novel geometric insight or proof required—standard textbook application of hyperbola formulas. |
| 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 |
| \(\frac{12^2}{a^2}-\frac{5^2}{b^2}=1\) and \(b^2 = a^2\left(\left(\frac{\sqrt{21}}{4}\right)^2-1\right)\) | M1 | Substitutes given point with 12 and 5 correctly positioned and substitutes given value of \(e\) into the correct eccentricity equation |
| \(b^2 = \frac{5}{16}a^2 \Rightarrow \frac{144}{a^2}-\frac{80}{a^2}=1 \Rightarrow a\) or \(a^2 = \ldots\) or \(\frac{45}{b^2}-\frac{25}{b^2}=0 \Rightarrow b\) or \(b^2 = \ldots\) | M1 | Solves simultaneously to obtain a value for \(a\) or \(a^2\) or \(b\) or \(b^2\) |
| \(a = 8,\ b = \sqrt{20}\) | A1, A1 | Allow equivalents for \(\sqrt{20}\) e.g. \(2\sqrt{5}\) or awrt 4.47. Do not allow \(\pm\) in either case |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(12 = a\sec\theta\), \(5 = b\tan\theta\) and \(b^2 = a^2\left(\left(\frac{\sqrt{21}}{4}\right)^2-1\right)\); \(\frac{b}{a}=\frac{\sqrt{5}}{4}\), \(\frac{b}{a}=\frac{5}{12}\csc\theta \Rightarrow \csc\theta = \frac{5}{3\sqrt{5}}\) | M1 | Substitutes given value of \(e\) into correct eccentricity equation and substitutes given point into correct parametric form and eliminates \(a\) and \(b\) |
| \(\csc\theta = \frac{5}{3\sqrt{5}} \Rightarrow a = \ldots\) or \(b = \ldots\) | M1 | Solves to obtain a value for \(a\) or \(b\) |
| \(a=8,\ b=\sqrt{20}\) | A1, A1 | Allow equivalents for \(\sqrt{20}\). Do not allow \(\pm\) in either case |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((\pm ae, 0) = (\pm 2\sqrt{21}, 0)\) | B1ft | Both (follow through on their \(a\)). Must be coordinates. |
# Question 2:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{12^2}{a^2}-\frac{5^2}{b^2}=1$ and $b^2 = a^2\left(\left(\frac{\sqrt{21}}{4}\right)^2-1\right)$ | M1 | Substitutes given point with 12 and 5 correctly positioned and substitutes given value of $e$ into the correct eccentricity equation |
| $b^2 = \frac{5}{16}a^2 \Rightarrow \frac{144}{a^2}-\frac{80}{a^2}=1 \Rightarrow a$ or $a^2 = \ldots$ or $\frac{45}{b^2}-\frac{25}{b^2}=0 \Rightarrow b$ or $b^2 = \ldots$ | M1 | Solves simultaneously to obtain a value for $a$ or $a^2$ or $b$ or $b^2$ |
| $a = 8,\ b = \sqrt{20}$ | A1, A1 | Allow equivalents for $\sqrt{20}$ e.g. $2\sqrt{5}$ or awrt 4.47. **Do not allow $\pm$ in either case** |
**Alternative to (a):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $12 = a\sec\theta$, $5 = b\tan\theta$ and $b^2 = a^2\left(\left(\frac{\sqrt{21}}{4}\right)^2-1\right)$; $\frac{b}{a}=\frac{\sqrt{5}}{4}$, $\frac{b}{a}=\frac{5}{12}\csc\theta \Rightarrow \csc\theta = \frac{5}{3\sqrt{5}}$ | M1 | Substitutes given value of $e$ into correct eccentricity equation and substitutes given point into correct parametric form and eliminates $a$ and $b$ |
| $\csc\theta = \frac{5}{3\sqrt{5}} \Rightarrow a = \ldots$ or $b = \ldots$ | M1 | Solves to obtain a value for $a$ or $b$ |
| $a=8,\ b=\sqrt{20}$ | A1, A1 | Allow equivalents for $\sqrt{20}$. **Do not allow $\pm$ in either case** |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\pm ae, 0) = (\pm 2\sqrt{21}, 0)$ | B1ft | Both (follow through on their $a$). Must be coordinates. |
**Total: 5 marks**
---2. The hyperbola $H$ has equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$
where $a$ and $b$ are positive constants.\\
The hyperbola $H$ has eccentricity $\frac { \sqrt { 21 } } { 4 }$ and passes through the point (12, 5).\\
Find
\begin{enumerate}[label=(\alph*)]
\item the value of $a$ and the value of $b$,
\item the coordinates of the foci of $H$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F3 2015 Q2 [5]}}