Parabola

A parabola is the locus of a point equidistant from a fixed point (focus) and a fixed line (directrix). This definition directly gives the equation: if the focus is (a, 0) and the directrix is x = -a, the standard parabola is $y^{2}$ = 4ax with vertex at the origin, axis along the x-axis, and the parabola opening rightward.

The four standard orientations are $y^{2}$ = 4ax (right), $y^{2}$ = -4ax (left), $x^{2}$ = 4ay (upward), and $x^{2}$ = -4ay (downward). The parameter 'a' is the distance from vertex to focus and also from vertex to directrix. The latus rectum is the chord through the focus perpendicular to the axis, with length 4a. The semi-latus rectum is 2a.

Any point on $y^{2}$ = 4ax can be parametrically represented as ($at^{2}$, 2at) where t is the parameter. This parametric form simplifies most computations dramatically.
The tangent at parameter t is ty = x + $at^{2}$, and the normal is y + tx = 2at + $at^{3}$.

For a chord joining points t1 and t2, the slope is 2/(t1 + t2). A focal chord (passing through the focus) satisfies t1*t2 = -1. The length of a focal chord with endpoints at parameters t1 and t2 is a(t1 - t2)² = a(t1 + t2)² + 4a. The minimum focal chord length is the latus rectum (4a).

The tangent at (x1, y1) on $y^{2}$ = 4ax is yy1 = 2a(x + x1) — the T = 0 formula. The tangent with slope m is y = mx + a/m. The condition for y = mx + c to be tangent is c = a/m.

The normal at ($at^{2}$, 2at) is y + tx = 2at + $at^{3}$. At most three normals can be drawn from an external point to a parabola. The feet of these normals satisfy a cubic equation, and their parameters t1, t2, t3 satisfy t1 + t2 + t3 = 0.

Reflection property: a ray parallel to the axis reflects through the focus after hitting the parabola. This is why satellite dishes and car headlights use parabolic shapes.

The key problem-solving concept is the parametric approach — represent points on the parabola as ($at^{2}$, 2at) and use parametric tangent/normal equations to avoid messy algebra.