Hyperbola

A hyperbola is the locus of a point whose difference of distances from two fixed points (foci) is constant: |PF₁ - PF₂| = 2a.
The standard equation is $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1, with foci at (+/-c, 0) where $c^{2}$ = $a^{2}$ + $b^{2}$ (contrast with ellipse: $c^{2}$ = $a^{2}$ - $b^{2}$) and eccentricity e = c/a > 1.

The transverse axis has length 2a along the x-axis, and the conjugate axis has length 2b along the y-axis. The vertices are (+/-a, 0). Unlike the ellipse, a need not be greater than b. The latus rectum has length $2b^{2}$/a. The semi-latus rectum l = $b^{2}$/a.

The asymptotes y = +/-(b/a)x are the key distinguishing feature. The hyperbola approaches these lines at infinity but never meets them.
The equation of asymptotes is $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 0.
The conjugate hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = -1 shares the same asymptotes.

The parametric representation is (asec($\theta$), btan($\theta$)), where $\theta$ is the eccentric angle.
The tangent at (x1, y1) is xx1/$a^{2}$ - yy1/$b^{2}$ = 1 (T = 0 formula).
The tangent with slope m is y = mx +/- $\sqrt{a^{2}*m^{2} - b^{2}}$, with tangency condition $c^{2}$ = $a^{2}$*$m^{2}$ - $b^{2}$ (note the minus sign, unlike the ellipse).

The normal at (x1, y1) is $a^{2}$*x/x1 + $b^{2}$*y/y1 = $a^{2}$ + $b^{2}$ = $c^{2}$. At most 4 normals can be drawn from an external point.

The auxiliary circle is $x^{2}$ + $y^{2}$ = $a^{2}$.
The director circle is $x^{2}$ + $y^{2}$ = $a^{2}$ - $b^{2}$, which exists only when a > b (eccentricity < $\sqrt{2}$).
When a = b (rectangular hyperbola, e = $\sqrt{2}$), the director circle reduces to a point. When a < b, no director circle exists (perpendicular tangents do not exist).

Key relationships: $b^{2}$ = $a^{2}$($e^{2}$ - 1), |SP - SP'| = 2a for any point P on the hyperbola, and the rectangular hyperbola xy = $c^{2}$ has eccentricity $\sqrt{2}$.