For what value of $c$ will the circle with equation $x^2 - 10x + y^2 + 10y + c = 0$ have a radius of length 1?
Complete the square on x, y
x^2 -10x + 25 + y^2 + 10y + 25 = 25 + 25 - c
(x -5)^2 + ( y+ 5)^2 = 50 - c
We want 50 - c =1
c = 49