Question 1205643
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Points that are 8 units from (-2,3) are on the circle {{{(x+2)^2+(y-3)^2 = 64}}}


Solve x+y = 0 for y to get y = -x


Plug y = -x into {{{(x+2)^2+(y-3)^2 = 64}}} to get {{{(x+2)^2+(-x-3)^2 = 64}}}


Do a bit of algebra to get the following
{{{(x+2)^2+(-x-3)^2 = 64}}}


{{{x^2+4x+4+x^2+6x+9 = 64}}}


{{{x^2+4x+4+x^2+6x+9-64 = 0}}}


{{{2x^2+10x-51 = 0}}}


Now apply the quadratic formula with: a = 2, b = 10, c = -51


{{{x = (-b+-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-10+-sqrt((10)^2-4(2)(-51)))/(2(2))}}}


{{{x = (-10+-sqrt(100 + 408))/(4)}}}


{{{x = (-10+-sqrt(508))/(4)}}}


{{{x = (-10+-  22.538855)/(4)}}}


{{{x = (-10+22.538855)/(4)}}} or {{{x = (-10-22.538855)/(4)}}}


{{{x = (12.538855)/(4)}}} or  {{{x = (-32.538855)/(4)}}}


{{{x = 3.134714}}} or  {{{x = -8.134714}}}
Each decimal value is approximate.


Each of those x values is plugged into y = -x to find their corresponding paired y value.


The two answer points are approximately <font color=red>(3.134714, -3.134714)</font> and <font color=red>(-8.134714, 8.134714)</font> which I've marked as points A and B respectively.


{{{
drawing(500,500,-10,10,-7,13,
graph(500,500,-10,10,-7,13,-100,-x),

circle(-2,3,8),

circle(3.134714,-3.134714,0.05),circle(3.134714,-3.134714,0.07),circle(3.134714,-3.134714,0.09),circle(3.134714,-3.134714,0.11),circle(3.134714,-3.134714,0.13),circle(3.134714,-3.134714,0.15),circle(-8.134714,8.134714,0.05),circle(-8.134714,8.134714,0.07),circle(-8.134714,8.134714,0.09),circle(-8.134714,8.134714,0.11),circle(-8.134714,8.134714,0.13),circle(-8.134714,8.134714,0.15),

circle(-2,3,0.05),circle(-2,3,0.07),circle(-2,3,0.09),circle(-2,3,0.11),circle(-2,3,0.13),circle(-2,3,0.15),

locate(3.134714,-3.134714-0.2,"A"),
locate(-8.134714,8.134714-0.2,"B"),
locate(-2,3-0.2,"C")

)
}}}
Point A is located at roughly (3.134714, -3.134714)
Point B is located at roughly (-8.134714, 8.134714)
Point C is the center of the circle which is (-2,3)
AC = 8 and BC = 8 since any point on the circle is 8 units from point C.
Desmos and GeoGebra are two graphing tools I recommend.
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