You need to determine the coordinates of the point` A` , hence, you need to use the information provided by the problem that the point `A` lies on the line `x + y + m = 0` .
Since the point `A` lies of the given line, you need to plug its coordinates into the line equation such that:
`x_A + y_A + m = 0 => m^2 + m + m = 0 => m^2 + 2m = 0`
Factoring out `m` , yields:
`m(m + 2) = 0`
Using the zero product rule yields:
`m = 0`
`m + 2 = 0 => m = -2`
Hence, evaluating m yields that the line `x + y = 0` passes through the origin `(0,0)` and the line `x + y - 2 = 0` passes through the point `(4,-2)` .
y = -x-m
A has the coordinates x=m^2, y=m
m = -m^2 -m
the solutions are m1=0 and m2 =-2
therefore the point A is (0,0) or A is (4,-2).