| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Hyperbola tangent/normal equation derivation |
| Difficulty | Challenging +1.3 Part (a) is a standard parametric differentiation exercise requiring implicit differentiation and point-normal form—routine for FP3 students. Part (b) involves more steps: finding where the normal meets the x-axis, using the eccentricity relationship, and solving a trigonometric equation, but follows a clear algorithmic path without requiring novel geometric insight. The 14-mark allocation and multi-step nature elevate it slightly above average difficulty. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03f Circle properties: angles, chords, tangents1.07s Parametric and implicit differentiation |
The hyperbola $C$ has equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.
\begin{enumerate}[label=(\alph*)]
\item Show that an equation of the normal to $C$ at the point $P(a \sec t, b \tan t)$ is
$$ax \sin t + by = (a^2 + b^2) \tan t.$$ [6]
\end{enumerate}
The normal to $C$ at $P$ cuts the $x$-axis at the point $A$ and $S$ is a focus of $C$. Given that the eccentricity of $C$ is $\frac{3}{2}$, and that $OA = 3OS$, where $O$ is the origin,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item determine the possible values of $t$, for $0 \leq t < 2\pi$. [8]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q16 [14]}}