2003 Paper 3 Q4

Year: 2003
Paper: 3
Question Number: 4

Course: LFM Pure and Mechanics
Section: Parametric equations

Difficulty: 1700.0 Banger: 1516.0

parametric equations tangent recurrence relation trigonometric periodic sequence curve tan identity algebraic manipulation

Problem

A curve is defined parametrically by \[ x=t^2 \;, \ \ \ y=t (1 + t^2 ) \;. \] The tangent at the point with parameter \(t\), where \(t\ne0\,\), meets the curve again at the point with parameter \(T\), where \(T\ne t\,\). Show that \[ T = \frac{1 - t^2 }{2t} \mbox { \ \ \ and \ \ \ } 3t^2\ne 1\;. \] Given a point \(P_0\,\) on the curve, with parameter \(t_0\,\), a sequence of points \(P_0 \, , \; P_1 \, , \; P_2 \, , \ldots\) on the curve is constructed such that the tangent at \(P_i\) meets the curve again at \(P_{i+1}\). If \(t_0 = \tan \frac{ 7 } {18}\pi\,\), show that \(P_3 = P_0\) but \(P_1\ne P_0\,\). Find a second value of \(t_0\,\), with \(t_0>0\,\), for which \(P_3 = P_0\) but \(P_1\ne P_0\,\).

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Difficulty Rating: 1700.0

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Banger Rating: 1516.0

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Problem source

A curve is defined parametrically by   
\[   
x=t^2 \;, \ \ \    
y=t (1 + t^2 )   
\;.   
\]   
The tangent at the  point with parameter $t$, where $t\ne0\,$, meets the    
curve again at the point with parameter $T$, where $T\ne t\,$. Show that   
\[   
T = \frac{1 - t^2 }{2t} \mbox { \ \ \ and \ \ \ } 3t^2\ne 1\;.   
\]   
Given a point $P_0\,$ on the curve, with parameter $t_0\,$,    
a sequence of points $P_0 \, , \; P_1 \, , \; P_2 \, , \ldots$    
on the curve is constructed such that the tangent at $P_i$ meets    
the curve again at $P_{i+1}$. If $t_0 = \tan \frac{ 7 } {18}\pi\,$,    
show that $P_3 = P_0$ but $P_1\ne P_0\,$.    
Find a second value of $t_0\,$, with $t_0>0\,$,   
for which $P_3 = P_0$ but  $P_1\ne P_0\,$.

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