On the circle …, Example 5 of 13.6 minimized

On the circle $g=x^2+y^2=1$, Example 5 of 13.6 minimized $f=xy-x-y$. (a) Set up the three Lagrange equations for $x,y,\lambda$. (b) The first two equations give $x=y=$______.
(c) There is another solution for the special value $\lambda =-\frac{1}{2}$, when the equations become ______. This is easy to miss but it gives $f_{\min }=-1$ at the point ______.

This exercise builds on another exercise in its source book, which isn't shown here.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §13.7 Constraints and Lagrange Multipliers, #14