Ellipse

An ellipse is the locus of a point whose sum of distances from two fixed points (foci) is constant and greater than the distance between the foci.
The standard equation is $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 (a > b > 0), with foci at (+/-c, 0) where $c^{2}$ = $a^{2}$ - $b^{2}$ and eccentricity e = c/a < 1.

The major axis has length 2a along the x-axis, the minor axis 2b along the y-axis. The vertices are (+/-a, 0) and the co-vertices are (0, +/-b). The latus rectum is the chord through a focus perpendicular to the major axis, with length $2b^{2}$/a. The semi-latus rectum l = $b^{2}$/a.

The parametric representation is (acos($\theta$), bsin($\theta$)), where $\theta$ is the eccentric angle (not the geometric angle at centre).
The tangent at eccentric angle $\theta$ is (xcos($\theta$))/a + (ysin($\theta$))/b = 1.

The tangent at point (x1, y1) on the ellipse is xx1/$a^{2}$ + yy1/$b^{2}$ = 1 (T = 0 formula).
The tangent with slope m is y = mx +/- $\sqrt{a^{2}*m^{2} + b^{2}}$.
The condition for y = mx + c to be tangent is $c^{2}$ = $a^{2}$*$m^{2}$ + $b^{2}$.

The normal at (x1, y1) is $a^{2}$*x/x1 - $b^{2}$*y/y1 = $a^{2}$ - $b^{2}$ = $c^{2}$. At most four normals can be drawn from an external point.

The auxiliary circle is $x^{2}$ + $y^{2}$ = $a^{2}$ (circumscribing the ellipse).
The director circle is $x^{2}$ + $y^{2}$ = $a^{2}$ + $b^{2}$ (locus of intersection of perpendicular tangents).

Key relationships: $b^{2}$ = $a^{2}$(1 - $e^{2}$), sum of focal distances SP + SP' = 2a for any point P on the ellipse, and the product of perpendicular distances from foci to any tangent equals $b^{2}$.

The key problem-solving concept is recognizing the appropriate form (point, slope, or parametric) for tangent/normal equations and leveraging the focal distance sum property SP + SP' = 2a.