The circle x² + y² = 8 is tangent to the line y = x + k for a positive value of k. What is that value of k?

A line is tangent to a circle when they share exactly one point. Substitute $y = x + k$ into $x^2 + y^2 = 8$: $x^2 + (x + k)^2 = 8$, which gives $2x^2 + 2kx + k^2 - 8 = 0$. One solution requires a zero discriminant: $(2k)^2 - 4(2)(k^2 - 8) = 4k^2 - 8k^2 + 64 = 0$, so $k^2 = 16$ and $k = 4$ for the positive value. Geometric check: the radius is $2\sqrt{2}$, and the distance from the origin to the line $x - y + k = 0$ is $\frac{k}{\sqrt{2}} = 2\sqrt{2}$, so $k = 4$. A wrong answer is $2\sqrt{2}$, the radius, which confuses distance with the intercept. Another is 16, which is $k^2$.