2026 NJC Promo Q9

In an interstellar game modelled on an \(xy\)-plane, the path of an asteroid is given by a curve with the equation

\[x=\sec\theta+a,\quad y=\tan\theta+b,\]

where \(\dfrac{\pi}{2}<\theta<\dfrac{3\pi}{2}\), \(a\le0\), \(b>0\).

1. Find the range of values of \(x\) in terms of \(a\).[2]

A measuring station is situated at origin \(O\). Let \(S\) be the distance between the asteroid and the measuring station.

2. Find \(S^2\) in terms of \(a\), \(b\) and \(\theta\).[1]
3. Given that \(\theta=\theta_1\) is the value of \(\theta\) which gives the minimum value of \(S\), show that \(\theta_1\) satisfies the equation \(2\tan\theta\sec\theta+a\tan\theta+b\sec\theta=0\).[2]

For the rest of the question, consider a simplified version of the game with \(a=0\).

4. Find the coordinates of the asteroid, in terms of \(b\), when it is the nearest to the measuring station along its path. You do not need to show that the distance between the asteroid and measuring station is a minimum at these coordinates.[3]
5. Show that the Cartesian equation of the asteroid's path is \(x^2-(y-b)^2=1\).[1]

As the asteroid travels, \(x\to-\infty\) and \(y\to\infty\), and the path of the asteroid approaches line \(l\).

6. Describe the transformation that would transform \(x^2-y^2=1\) onto \(x^2-(y-b)^2=1\) and deduce the equation of \(l\).[2]