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For what positive value of c does the line y = -x + c intersect the circle x^2 + y^2 = 1 in exactly one point?

Sub   -x + c  in for y in the circular equation and we have that

x^2  +  (-x + c)^2  = 1   simplify

x^2 + c^2 - 2cx + x^2 = 1

2x^2 - 2cx +  (c^2 - 1)  = 0

If these intersect in exactly one point, the discriminant must  = 0....so....

(2c)^2  - 4(2)(c^2 - 1)  = 0

4c^2 - 8c^2 + 8  = 0

-4c^2 + 8  = 0      subtract 8 from both sides

-4c^2  = -8            divide both sides by -4

c^2  = 2                 take the positive root

c  = √2

Here's a graph : https://www.desmos.com/calculator/lithgiri0g