Analytic Geometry

Core Theoretical Principles

Analytic geometry bridges algebraic formulation with geometric intuition. Our lectures focus heavily on conic sections, parametric curves, and moving locus trajectories.

1. Standard Parabola Equations

A standard parabola with a vertex at the origin $(0, 0)$ and focal parameter $p > 0$:

$$ y^2 = 4px \quad (\text{Horizontal Axis}) \qquad \text{or} \qquad x^2 = 4py \quad (\text{Vertical Axis}) $$

The directrix is given by the linear equation $x = -p$, and the coordinates of the focus are $F(p, 0)$. For any point $P(x, y)$ on the conic, the distance to the focus equals the perpendicular distance to the directrix:

$$ d(P, F) = d(P, \text{directrix}) \implies \sqrt{(x - p)^2 + y^2} = |x + p| $$

2. Tangent Line to a Conic Section

Using the method of split coordinates, the equation of the tangent line to the parabola $y^2 = 4px$ at a given point $P_0(x_0, y_0)$ is:

$$ y \cdot y_0 = 2p(x + x_0) $$

For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the tangent line at $P_0(x_0, y_0)$ takes the symmetric form:

$$ \frac{x \cdot x_0}{a^2} + \frac{y \cdot y_0}{b^2} = 1 $$

3. Moving Locus & Triangle Trajectories

When points move dynamically along constraints (e.g. vertices of a triangle traversing intersecting lines), their barycenters or circumcenters trace out secondary conic loci. Watch our step-by-step video proofs below to see these geometric transformations visualized in real time.