Question 3 - A-Level Maths

Edexcel FP1 2014 January — Question 3 4 marks

Exam Board	Edexcel
Module	FP1 (Further Pure Mathematics 1)
Year	2014
Session	January
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Parametric curves and Cartesian conversion
Type	Find intersection points
Difficulty	Standard +0.3 This is a straightforward FP1 parametric question requiring students to find coordinates of two points, calculate the gradient of the chord PQ, then use perpendicular gradient properties. All steps are routine applications of standard techniques with no conceptual challenges beyond basic coordinate geometry.
Spec	1.03b Straight lines: parallel and perpendicular relationships 1.03g Parametric equations: of curves and conversion to cartesian

3. A rectangular hyperbola has parametric equations $$x = 2 t , \quad y = \frac { 2 } { t } , \quad t \neq 0$$ Points $P$ and $Q$ on this hyperbola have parameters $t = \frac { 1 } { 2 }$ and $t = 4$ respectively.
The line $L$, which passes through the origin $O$, is perpendicular to the chord $P Q$.
Find an equation for $L$.

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Answer	Marks	Guidance
$x = 2t, y = \frac{2}{t}, t \ne 0$
$t = \frac{1}{2} \Rightarrow P(1, 4)$, $t = 4 \Rightarrow Q\left(8, \frac{1}{2}\right)$	B1	Coordinates for either $P$ or $Q$ are correctly stated. (Can be implied)
$m(PQ) = \frac{\frac{1}{2} - 4}{8 - 1} = \left\{= -\frac{1}{2}\right\}$	M1	An attempt to find the gradient of the chord $PQ$
$m(L) = 2$	M1	Applying $m(L) = \frac{-1}{\text{their } m(PQ)}$
So, $L: y = 2x$	A1 oe

| $x = 2t, y = \frac{2}{t}, t \ne 0$ | | |
| $t = \frac{1}{2} \Rightarrow P(1, 4)$, $t = 4 \Rightarrow Q\left(8, \frac{1}{2}\right)$ | B1 | Coordinates for either $P$ or $Q$ are correctly stated. (Can be implied) |
| $m(PQ) = \frac{\frac{1}{2} - 4}{8 - 1} = \left\{= -\frac{1}{2}\right\}$ | M1 | An attempt to find the gradient of the chord $PQ$ |
| $m(L) = 2$ | M1 | Applying $m(L) = \frac{-1}{\text{their } m(PQ)}$ |
| So, $L: y = 2x$ | A1 oe | |

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3. A rectangular hyperbola has parametric equations

$$x = 2 t , \quad y = \frac { 2 } { t } , \quad t \neq 0$$

Points $P$ and $Q$ on this hyperbola have parameters $t = \frac { 1 } { 2 }$ and $t = 4$ respectively.\\
The line $L$, which passes through the origin $O$, is perpendicular to the chord $P Q$.\\
Find an equation for $L$.\\

\hfill \mbox{\textit{Edexcel FP1 2014 Q3 [4]}}

This paper (10 questions)

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Q1 5 Q2 7 Q3 4 Q4 5 Q5 8 Q6 9 Q7 6 Q8 12 Q9 8 Q10 11

Answer	Marks	Guidance
\(x = 2t, y = \frac{2}{t}, t \ne 0\)
\(t = \frac{1}{2} \Rightarrow P(1, 4)\), \(t = 4 \Rightarrow Q\left(8, \frac{1}{2}\right)\)	B1	Coordinates for either \(P\) or \(Q\) are correctly stated. (Can be implied)
\(m(PQ) = \frac{\frac{1}{2} - 4}{8 - 1} = \left\{= -\frac{1}{2}\right\}\)	M1	An attempt to find the gradient of the chord \(PQ\)
\(m(L) = 2\)	M1	Applying \(m(L) = \frac{-1}{\text{their } m(PQ)}\)
So, \(L: y = 2x\)	A1 oe