2009 ACJC P1 Q7

A point \(\left( x,y \right)\) lies on the curve with equation \(\frac{1}{x}+\frac{1}{y}=\frac{1}{a}\), where \(a\) is a non-zero constant, \(x\ne 0\) and \(y\ne 0\).

1. Express \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(x\) and \(y\), and explain clearly whether \(y\) is an increasing or a decreasing function.[3]
2. State, giving a reason, whether there are any stationary points on this curve.[2]
3. Find the equation of the tangent to the curve at the point \(\left( 2a,2a \right)\) and determine if the tangent cuts the curve again.[5]