Mathematics

A parabola is defined by a single geometric property: every point on it is equidistant from a fixed point (the focus) and a fixed line (the directrix). Start by using this distance definition—set the distance from a point $(x, y)$ to the focus $(p, 0)$ equal to its distance to the line $x = -p$—then simplify algebraically to see if you recover the given equation. This reverses the usual direction and proves the equation captures exactly this parabolic property.

A parabola is defined by a single geometric property: every point on it is equidistant from a fixed point (the focus) and a fixed line (the directrix). Start by using this distance definition—set the distance from a point $(x, y)$ to the focus $(p, 0)$ equal to its distance to the line $x = -p$—then simplify algebraically to see if you recover the given equation. This reverses the usual direction and proves the equation captures exactly this parabolic property.