The Hyperbola

1. Core Concept Overview & Intuition

A hyperbola is a curve defined as a locus of points. The approved notes develop the standard equation by considering a fixed point $F$ (the focus) and a fixed line $D$ (the directrix). The hyperbola is the set of points whose distance from the focus bears a constant ratio to its perpendicular distance from the directrix, where that constant ratio (the eccentricity) is greater than one. This is the geometric definition from which the standard equation is derived.

The curve possesses two lines of symmetry, the $x$-axis and the $y$-axis (for a hyperbola centered at the origin). It consists of two separate branches opening in opposite directions. The segment of the axis between the two vertices is the transverse axis, and the segment perpendicular to it through the center is the conjugate axis.

An alternative and equivalent definition, stated in the approved notes, is that a hyperbola is the locus of a point that moves such that the difference between its distances from two fixed points (the foci) is a constant equal to $2a$. This constant-difference property is the defining metric feature of the hyperbola, just as a constant sum of distances defines the ellipse.

The hyperbola also possesses asymptotes: lines that the branches approach ever more closely as they extend outward. When the transverse and conjugate axes are equal in length, the hyperbola is called a rectangular hyperbola, and its asymptotes are perpendicular.

2. Mathematical Formalization & Core Definitions

2.1 Standard Equation and Geometric Definition

The standard equation is developed from a fixed focus $F$ and a fixed directrix $D$. Let the axis of the hyperbola be the horizontal line perpendicular to the directrix through the focus, and let $V$ be the point of intersection of the hyperbola and the axis. The derivation leads to the standard equation

$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1,\qquad b^{2}=c^{2}-a^{2},
$$

where $a$ and $b$ are positive constants and $c$ is the focal distance from the center. The curve has two lines of symmetry, the $x$-axis and the $y$-axis. The quantity $2a$ is the length of the transverse axis and $2b$ is the length of the conjugate axis.

The constant-difference property gives the alternative definition: the difference between the distances from the two fixed foci is a constant equal to $2a$.

2.2 Transverse and Conjugate Axes, Vertices, and Foci

For the standard hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$:

• The transverse axis has length $2a$; the conjugate axis has length $2b$.
• The vertices lie on the transverse axis at $(\pm a,0)$.
• The foci lie on the transverse axis at $(\pm c,0)$, where $c^{2}=a^{2}+b^{2}$.

2.3 Directrices, Eccentricity, and Latus Rectum

The eccentricity is denoted $e$ and satisfies $e=\frac{c}{a}$, with $e>1$ for a hyperbola. The directrices are the lines $x=\pm\frac{a}{e}$.

The latus rectum is the chord through a focus perpendicular to the transverse axis. Its length is obtained by letting the half-length be $\ell$, where $\ell$ is the distance from the focus along the latus rectum to the curve. Substitution into the standard equation gives the length of the latus rectum as $\frac{2b^{2}}{a}$.

2.4 Asymptotes and Rectangular Hyperbolas

The asymptotes of the standard hyperbola are the lines

$$
y=\pm\frac{b}{a}x.
$$

A rectangular hyperbola is one for which the transverse and conjugate axes are equal, so that $a=b$; its asymptotes are then perpendicular.

2.5 Translated Hyperbolas with Center $(h,k)$

If the center of the hyperbola is shifted to the position $(h,k)$, the equation becomes

$$
\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1.
$$

The vertices, foci, and asymptotes are correspondingly translated: vertices at $(h\pm a,k)$, foci at $(h\pm c,k)$, and asymptotes $y-k=\pm\frac{b}{a}(x-h)$.

2.6 Equations of Tangent and Normal

The tangent to a curve at a point is the line touching the curve there; the normal is the line perpendicular to the tangent at that point. The approved notes state that the normal at any point on the curve can be easily obtained since it is the line perpendicular to the tangent at that point. Thus, once the tangent is determined, the normal follows from the perpendicularity of their slopes.