1998 Paper 2 Q6

Year: 1998
Paper: 2
Question Number: 6

Course: LFM Pure and Mechanics
Section: Parametric equations

Difficulty: 1600.0 Banger: 1500.0

parametric equations cycloid tangent-normal curve sketching gradient differentiation parametric differentiation

Problem

Two curves are given parametrically by \[ x_{1}=(\theta+\sin\theta),\qquad y_{1}=(1+\cos\theta),\tag{1} \]and \[ x_{2}=(\theta-\sin\theta),\qquad y_{1}=-(1+\cos\theta),\tag{2} \] Find the gradients of the tangents to the curves at the points where \(\theta= \pi/2\) and \(\theta=3\pi/2\). Sketch, using the same axes, the curves for \(0\le\theta \le 2\pi\). Find the equation of the normal to the curve (1) at the point with parameter \(\theta\). Show that this normal is a tangent to the curve (2).

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Rating Information

Difficulty Rating: 1600.0

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Banger Rating: 1500.0

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Problem source

Two curves are given parametrically by
\[
x_{1}=(\theta+\sin\theta),\qquad y_{1}=(1+\cos\theta),\tag{1}
\]and
\[
x_{2}=(\theta-\sin\theta),\qquad y_{1}=-(1+\cos\theta),\tag{2}
\]
Find the gradients of the tangents to the curves at the points where 
$\theta= \pi/2$ and $\theta=3\pi/2$. 
Sketch, using the same axes, the
curves
for $0\le\theta \le 2\pi$.
Find the equation of the normal to the
curve (1) at the point with parameter $\theta$. Show that this normal is
a tangent to the curve (2).

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