| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Ellipse locus problems |
| Difficulty | Challenging +1.2 This is a structured multi-part FP1 conic sections question requiring standard techniques: eccentricity formula (recall), implicit differentiation for the normal (routine calculus), simultaneous equations for intersection, and verification that a locus is an ellipse. While it involves several steps and parametric reasoning, each part follows predictable methods without requiring novel insight. Slightly above average difficulty due to the algebraic manipulation and the locus verification in part (d). |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
\begin{enumerate}
\item The ellipse $E$ has equation
\end{enumerate}
$$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$$
(a) Determine the exact value of the eccentricity of $E$
The points $P ( 4 \cos \theta , 3 \sin \theta )$ and $Q ( 4 \cos \theta , - 3 \sin \theta )$ lie on $E$ where $0 < \theta < \frac { \pi } { 2 }$ The line $l _ { 1 }$ is the normal to $E$ at the point $P$
\begin{center}
\end{center}
(b) Use calculus to show that $l _ { 1 }$ has equation
$$4 x \sin \theta - 3 y \cos \theta = 7 \sin \theta \cos \theta$$
The line $l _ { 2 }$ passes through the origin and the point $Q$ The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $R$\\
(c) Determine, in simplest form, the coordinates of $R$\\
(d) Hence show that, as $\theta$ varies, $R$ lies on an ellipse which has the same eccentricity as ellipse $E$
\hfill \mbox{\textit{Edexcel FP1 2023 Q4}}