I am trying exercises from the book Apostol Modular Functions and Dirichlet series in number theory and I am struck on proving equality in this exercise of chapter 2 .
I am adding image of some previous problems also because they contain definations useful in problem 8 . But please note that I have doubt only in problem 8 .
Image - 
My attempt - Using result that fundamental region is $|\tau|>1$ and $|\tau +\tau^* |<1$ , where $\tau^*$ = conjugate of $\tau$ and defination of reduced form that a reduced form is one whose representative $\tau$ belongs to $R_\Gamma$ I obtained relation $b<|a|$. and using property of discriminant I obtained $|a|<c$.
My doubt -> I don't know how to obtain $b\le c$ in first part and how to obtain $b\le a=c$ in second part.
My attempt in these - when I put $a= b$ in conditions of 1st part I get real part of representative $= - 1/2$ , which is a contradiction to defination of fundamental region. In 2 nd part, when try to get something by putting $a=c$, I get $4a^2 > 4 a^2$ which implies $0>0$ .
Can someone please help in proving these Equalities in proof. Inequalities I have proved.