2025 DHS J1 WA2 Q3

The flight path (see diagram) of a paper plane, towards a wall, can be modelled by the equation \(y=\frac{x-1}{{{x}^{2}}}\); \(x > 0\). Its horizontal position, \(x\) m, is measured from the wall and its vertical position, \(y\) m, is measured from the ground.

1. Find the equation of the tangent to the curve at \(x=a\).
   Give your answer in the form \(y=mx+c\), where \(m\) and \(c\) are expressed in terms of \(a\).[2]

The line of sight from the tip of the paper plane at \(x=a\) is tangential to its flight path and meets the wall at \(B\left( 0,h \right)\).

2. Show that \(h=\frac{2}{a}-\frac{3}{{{a}^{2}}}\).[1]
3. As \(a\) varies, find the exact greatest value of \(h\) as the paper plane flies towards the wall.[3]