Curve formed using straight lines

Each blue line has an equation of $y = \cfrac{-\frac{n - k}{n}}{\frac{k}{n}}x + \cfrac{n - k}{n}$ or $y = x - \cfrac{nx}{k} + 1 - \cfrac{k}{n}$ .

The red curve is made up of maximum heights of all the blue lines at a given $x$ -coordinate, so $\cfrac{dy}{dk} = \cfrac{nx}{k^2} - \cfrac{1}{n} = 0$ , which solves to $k = \pm n \sqrt{x}$ .

Substituting $k = \pm n \sqrt{x}$ into $y = x - \cfrac{nx}{k} + 1 - \cfrac{k}{n}$ and rearranging gives $x^2 - 2xy + y^2 - 2x - 2y + 1 = 0$ .

This conic has a discriminant of $(-2)^2 - 4 \cdot 1 \cdot 1 = 0$ , which means the red curve is a parabola .