| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Conic tangent through external point |
| Difficulty | Challenging +1.2 This is a standard Further Maths conic sections question requiring algebraic manipulation to derive the intersection condition (part a), applying the tangent condition via discriminant (part b), and solving a system to find specific tangents (part c). While it involves multiple steps and Further Maths content, the techniques are routine for FP3 students—substitution, expanding, collecting terms, and using b²=a²m²-c². The final part requires solving simultaneous equations but follows a well-established method. More challenging than typical A-level Core questions due to the topic and algebraic complexity, but standard for this module. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02f Solve quadratic equations: including in a function of unknown1.02y Partial fractions: decompose rational functions |
6. The hyperbola $H$ has equation $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, where $a$ and $b$ are constants.
The line $L$ has equation $y = m x + c$, where $m$ and $c$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Given that $L$ and $H$ meet, show that the $x$-coordinates of the points of intersection are the roots of the equation
$$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$
Hence, given that $L$ is a tangent to $H$,
\item show that $a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }$.
The hyperbola $H ^ { \prime }$ has equation $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1$.
\item Find the equations of the tangents to $H ^ { \prime }$ which pass through the point $( 1,4 )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q6 [10]}}