A discriminant reveals a forbidden output interval

The curve $C$ has equation $y=\frac{2x^2+4}{3(x+\lambda+1)}$. It is given that $C$ has a vertical asymptote $x=0$.

(i) Determine the value of $\lambda$.1 mark

(ii) Find the equation of the other asymptote of $C$.2 marks

(iii) Prove, using an algebraic method, that $C$ cannot lie between $-\frac{4\sqrt a}{3}<y<\frac{4\sqrt a}{3}$, $a<0$, where $a$ is to be determined.3 marks

(iv) When $\lambda=0$, the graph of $y=\frac{2x^2+4}{3(x+\lambda+1)}+k$ cuts the $x$-axis at 2 distinct points where $k$ is a positive integer.

(a) Sketch the graph of $y=\frac{2x^2+4}{3(x+\lambda+1)}+k$ when $\lambda=0$, indicating clearly all asymptotes and the coordinates of the axial intercepts.4 marks

(b) Deduce the least possible integer value of $k$.1 mark

[2009/NYJC/Prelim/I/7]