Hyperbolas as Geometric Loci

Definition Let $F_1$ and $F_2$ be two distinct points of the plane, let $2c = \lvert F_1F_2 \rvert > 0$ be the distance between them, and let $a$ be a number with $0 < a < c$. The hyperbola with foci $F_1$ and $F_2$ is the set of all points $P$ of the plane with

The absolute value lets either focus be the farther one. The condition $0 < a < c$ is needed. By the triangle inequality the difference is never more than $2c$, and it equals $2c$ only on the two rays of the line $F_1F_2$ that start at the foci and point away from each other. A difference above $2c$ gives no points, and a difference of $0$ gives the perpendicular bisector of $F_1F_2$, a line.

Definition The foci of a hyperbola are the two fixed points $F_1$ and $F_2$ of its definition, and the line through them is the focal axis. The center is the midpoint of $F_1F_2$, so each focus lies at distance $c$ from the center. The hyperbola falls into two separate pieces called branches. On the branch around $F_2$ the condition reads $\lvert PF_1 \rvert - \lvert PF_2 \rvert = 2a$, and on the branch around $F_1$ it reads $\lvert PF_2 \rvert - \lvert PF_1 \rvert = 2a$. Each branch bends around the focus nearer to it, and the whole curve is symmetric about its center.

Definition The vertices of a hyperbola are the points where it meets its focal axis. There are exactly two, they lie between the foci, and each is at distance $a$ from the center. The segment between them has length $2a$. For the hyperbola with foci $(\pm c, 0)$ the vertices are $(\pm a, 0)$.

Definition For a hyperbola with $0 < a < c$, define the positive number $b = \sqrt{c^2 - a^2}$, so that

With the center at the origin and the foci at $(\pm c, 0)$, a point $(x, y)$ lies on the hyperbola exactly when

The derivation is the same as for the ellipse, squaring twice, and it adds no extra points. Every point $P$ of this equation has $\lvert x \rvert \ge a$, and its distances to $F_1 = (-c, 0)$ and $F_2 = (c, 0)$ are

Since $\frac{c}{a}\lvert x \rvert \ge c > a$, both numbers inside the absolute values have the sign of $x$, so the two distances differ by exactly $2a$. With the foci at $(0, \pm c)$ the equation is

The foci lie on the axis of the variable with the plus sign. Unlike for the ellipse, $a$ may be smaller than $b$.

Definition The transverse axis of a hyperbola is the segment joining its two vertices. It lies on the focal axis and has length $2a$. The conjugate axis is the segment through the center, perpendicular to the transverse axis, of length $2b$. For the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ it runs from $(0, -b)$ to $(0, b)$. The curve never meets the line of the conjugate axis, because that line is the perpendicular bisector of $F_1F_2$ and each of its points is equally far from both foci.

Definition The asymptotes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ are the lines

On the part of the curve in the first quadrant, $y = \frac{b}{a}\sqrt{x^2 - a^2}$, and its vertical distance below the first line is

which is positive and tends to $0$ as $x \to \infty$. By symmetry each of the four ends of the curve approaches one of the two lines. For $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ the asymptotes are $y = \pm \frac{a}{b}x$. In both cases they are the lines through the diagonals of the rectangle centered at the center whose sides of length $2a$ run parallel to the transverse axis and whose sides of length $2b$ run parallel to the conjugate axis.

Definition The eccentricity of a hyperbola is the ratio

It fixes the ratio of the axes through

so for a hyperbola with foci on the $x$-axis the asymptotes have slopes $\pm\sqrt{e^2 - 1}$. When $a = b$ the eccentricity is $\sqrt{2}$ and the asymptotes are perpendicular.