| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Ellipse locus problems |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths ellipse question requiring knowledge of foci, directrices, and the focal distance property. Part (a) involves standard formulas, part (b) is a routine verification using the ellipse definition, and part (c) requires the chord midpoint locus technique (differentiation or parametric approach). While it's Further Maths content and multi-step, these are well-practiced techniques from FP3 with no novel insight required, making it moderately above average difficulty. |
| 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 |
| \(b^2 = a^2(1-e^2) \Rightarrow e^2 = \frac{3}{4}\) or \(e = \frac{\sqrt{3}}{2}\) | M1A1 | M1: Uses correct eccentricity formula. A1: \(e^2 = \frac{3}{4}\) or \(e = \frac{\sqrt{3}}{2}\) (allow \(e = \pm\frac{\sqrt{3}}{2}\)) |
| Foci: \((\pm ae, 0) \Rightarrow (\pm\sqrt{3}, 0)\) | B1 | Both correct as coordinates |
| Directrices: \(x = \pm\frac{a}{e} \Rightarrow x = \pm\frac{4}{\sqrt{3}}\) | B1 | Both directrices correct as equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(PF_1 = ePN_1\) and \(PF_2 = ePN_2\) | M1 | Use of definition of ellipse for either \(PF_1\) or \(PF_2\) |
| \(PF_1 + PF_2 = e(PN_1 + PN_2) = eN_1N_2\) | dM1A1 | dM1: (their \(e\))\(\times 2\)(their \(\frac{4}{\sqrt{3}}\)). Dependent on previous M. A1: \(\frac{\sqrt{3}}{2}\times\left(2\times\frac{4}{\sqrt{3}}\right)\) |
| \(= 4\) | A1 | cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{x^2}{4} + (mx+c)^2 = 1\) | M1 | Substitutes line with gradient \(m\) into ellipse equation |
| \((1+4m^2)x^2 + 8mcx + 4(c^2-1) = 0\) | A1 | Correct quadratic in \(x\) with terms collected |
| \(x = \frac{1}{2}(\text{sum of roots}) = \frac{-4mc}{1+4m^2}\) | M1 | Attempts \(\frac{1}{2}\)(sum of roots) |
| \(\Rightarrow c = -\frac{(1+4m^2)x}{4m}\) | A1 | Correct expression for \(c\) in terms of \(m\) and \(x\) |
| So \(y = mx - \frac{(1+4m^2)x}{4m} = -\frac{1}{4m}x\) | ddM1A1 | ddM1: Substitutes back into \(y = mx+c\). Depends on both previous M marks. A1: Correct equation |
# Question 8:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $b^2 = a^2(1-e^2) \Rightarrow e^2 = \frac{3}{4}$ or $e = \frac{\sqrt{3}}{2}$ | M1A1 | M1: Uses correct eccentricity formula. A1: $e^2 = \frac{3}{4}$ or $e = \frac{\sqrt{3}}{2}$ (allow $e = \pm\frac{\sqrt{3}}{2}$) |
| Foci: $(\pm ae, 0) \Rightarrow (\pm\sqrt{3}, 0)$ | B1 | Both correct as coordinates |
| Directrices: $x = \pm\frac{a}{e} \Rightarrow x = \pm\frac{4}{\sqrt{3}}$ | B1 | Both directrices correct as equations |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $PF_1 = ePN_1$ and $PF_2 = ePN_2$ | M1 | Use of definition of ellipse for either $PF_1$ or $PF_2$ |
| $PF_1 + PF_2 = e(PN_1 + PN_2) = eN_1N_2$ | dM1A1 | dM1: (their $e$)$\times 2$(their $\frac{4}{\sqrt{3}}$). Dependent on previous M. A1: $\frac{\sqrt{3}}{2}\times\left(2\times\frac{4}{\sqrt{3}}\right)$ |
| $= 4$** | A1** | cso |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{x^2}{4} + (mx+c)^2 = 1$ | M1 | Substitutes line with gradient $m$ into ellipse equation |
| $(1+4m^2)x^2 + 8mcx + 4(c^2-1) = 0$ | A1 | Correct quadratic in $x$ with terms collected |
| $x = \frac{1}{2}(\text{sum of roots}) = \frac{-4mc}{1+4m^2}$ | M1 | Attempts $\frac{1}{2}$(sum of roots) |
| $\Rightarrow c = -\frac{(1+4m^2)x}{4m}$ | A1 | Correct expression for $c$ in terms of $m$ and $x$ |
| So $y = mx - \frac{(1+4m^2)x}{4m} = -\frac{1}{4m}x$ | ddM1A1 | ddM1: Substitutes back into $y = mx+c$. Depends on both previous M marks. A1: Correct equation |\begin{enumerate}
\item The ellipse $E$ has equation $x ^ { 2 } + 4 y ^ { 2 } = 4$\\
(a) (i) Find the coordinates of the foci, $F _ { 1 }$ and $F _ { 2 }$, of $E$.\\
(ii) Write down the equations of the directrices of $E$.\\
(b) Given that the point $P$ lies on the ellipse, show that
\end{enumerate}
$$\left| P F _ { 1 } \right| + \left| P F _ { 2 } \right| = 4$$
A chord of an ellipse is a line segment joining two points on the ellipse.\\
The set of midpoints of the parallel chords of $E$ with gradient $m$, where $m$ is a constant, lie on a straight line $l$.\\
(c) Find an equation of $l$.\\
\hfill \mbox{\textit{Edexcel FP3 2015 Q8 [14]}}