Answer
$(\frac{-5+\sqrt {127}}{2}, \frac{5-\sqrt {127}}{2})$ and $(\frac{-5-\sqrt {127}}{2},\frac{5+\sqrt {127}}{2})$
Work Step by Step
Step 1. Points that are 8 units from $(-2,3)$ are on a circle: $(x+2)^2+(y-3)^2=64$
Step 2. Let $y=-x$ in the equation, we have $(x+2)^2+(-x-3)^2=64$ or $2x^2+10x-51=0$ which gives $x=\frac{-10\pm\sqrt {100+4(2)(51)}}{2(2)}=\frac{-10\pm\sqrt {508}}{4}=\frac{-5\pm\sqrt {127}}{2}$
Step 3. The points are $(\frac{-5+\sqrt {127}}{2}, \frac{5-\sqrt {127}}{2})$ and $(\frac{-5-\sqrt {127}}{2},\frac{5+\sqrt {127}}{2})$
