The circle x² + y² = 32 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 it meets the circle at exactly one point. Substitute $y = x + k$ into $x^2 + y^2 = 32$: $x^2 + (x + k)^2 = 32$, so $2x^2 + 2kx + k^2 - 32 = 0$. Setting the discriminant to zero: $(2k)^2 - 4(2)(k^2 - 32) = 4k^2 - 8k^2 + 256 = 0$, so $k^2 = 64$ and $k = 8$. Geometric check: the radius is $\sqrt{32} = 4\sqrt{2}$, and the distance from the origin to the line $x - y + k = 0$ is $\frac{k}{\sqrt{2}} = 4\sqrt{2}$, giving $k = 8$. A common wrong answer is 32, which is $r^2$ and not $k$.