9 Two astronomers wish to model the path of motion of a particle in two dimensions.
Experimental results show that the position of the particle can be found using the parametric equations
$$x = 2 \cos \theta - \sin \theta + 2 \quad y = \cos \theta + 2 \sin \theta - 1 \quad \left( 0 \leq \theta \leq 360 ^ { \circ } \right)$$
One astronomer uses trigonometry.
- Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants to be determined.
Show also that, for the same values of \(R\) and \(\alpha\),
$$\cos \theta + 2 \sin \theta = R \sin ( \theta + \alpha )$$
- Hence, or otherwise, show that the path of particle may be written in the form
$$( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 5$$
Describe the path of the particle.
The second astronomer sets up a first order differential equation with the condition that \(x = 4\) when \(y = 0\).
- Verify that the point with parameter \(\theta = 0\) has coordinates \(( 4,0 )\).
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Deduce that \(x\) and \(y\) satisfy the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x - 2 } { y + 1 }$$
- Solve this differential equation, using the condition that \(y = 0\) when \(x = 4\).
Hence show that the two solutions give the same cartesian equation for the path of particle.