Introduction to Conics and the Parabola

A conic section is the curve obtained where a plane cuts a right circular cone; depending on the angle of the cut, the resulting curve is a circle, an ellipse, a parabola, or a hyperbola. Algebraically, every conic is the locus of a point whose distance from a fixed point (the focus) bears a constant ratio — the eccentricity, $e$ — to its perpendicular distance from a fixed line (the directrix). This chapter's treatment of conics is introductory: the goal is to recognise the standard equation of each conic and read off its basic features, not to derive each curve from first principles.

The parabola is the conic with eccentricity $e = 1$ (distance from focus always exactly equals distance from directrix — precisely the locus condition seen in Section 2). Its simplest standard form, with vertex at the origin and axis along the $x$-axis, is:

For this standard parabola: the focus is at $(a, 0)$, the directrix is the line $x = -a$, and the length of the latus rectum (the chord through the focus, perpendicular to the axis) is $4a$ — found by substituting $x=a$ into the equation, giving $y = \pm 2a$, a chord of length $4a$. …