2024 NYJC Promo Q10

A tiny robot moves in a way that traces out the curve with parametric equations given by \[x = 1 + t^2,\quad y = 2 \sin^{-1} t\] from the point where \(t = -1\) to the point where \(t = 1\).

1. Sketch the curve traced out by the robot, indicating clearly the exact coordinates of the endpoints.[2]
2. Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(t\).[2]

The robot crosses the \(x\)-axis at the point \(P\).

3. Find the cartesian equation of the tangent to the curve at \(P\).[2]
4. Find the angle between the direction in which the robot is moving and the positive \(x\)-axis at the instant it reaches the point where \(y = \frac{\pi}{3}\).[2]

A tiny magnet is placed at the point \((2, 2)\).

5. Show that the distance \(s\) between the robot and the magnet is given by
   \(s = \sqrt{\left(t^2 - 1\right)^2 + 4\left(\sin^{-1} t - 1\right)^2}.\)[1]
6. The magnet will attract the robot if it comes within \(0.25\) unit of its position. By using differentiation to find the minimum distance between the magnet and the robot, determine whether the robot will be attracted by the magnet.
   [You need not show that the distance is a minimum.][5]