A hyperbola is the locus of points $P$ in the Euclidean plane such that the distances of $P$ from two fixed points (the foci $F_1$ and $F_2$ ) differ from each other by a constant amount ($\pm2a$ ). The line segments connecting a point of the hyperbola to the foci are called focal radii.

We obtain the simplest equation for the hyperbola by choosing the foci on the other coordinate axis and equidistant ($= c > a > 0$ ) from the origin. Let $F_1 = (-c,\,0)$ and $F_2 = (c,\,0)$ . Then the locus condition for the point $P = (x,\,y)$ of the hyperbola is $$\sqrt{(x+c)^2+y^2}-\sqrt{(x-c)^2+y^2} = \pm2a.$$ The simplifying of this, via two squarings, yields the equation of the hyperbola

(1)

Since the equation (1) contains only the squares of $x$ and $y$ we can infer that the hyperbola is symmetric with respect to the coordinate axes and the origin; this is naturally clear on grounds of the definition of hyperbola, too.

Solving (1) for $y$ we get

(2)

The line segment connecting the apices is the transversal axis of the hyperbola. The line segment on the $y$ -axis from $-b$ to $b$ is the conjugate axis of the hyperbola. By the Pythagorean theorem, the equation $b^2 = c^2-a^2$ shows that the distance between an end of the transversal and an end of the conjucate axis is equal to $c$ .

Let's consider the part $$y = \frac{b}{a}\sqrt{x^2-a^2}$$ of the hyperbola situated in the first quadrant ($x > a$ ) and the line $$y \;=\; \frac{b}{a}x.$$ The difference of their ordinates corresponding a same abscissa $x$ may be written $$\Delta \;:=\; \frac{b}{a}(x\!-\!\sqrt{x^2\!-\!a^2}) \;=\; \frac{ab}{x\!+\!\sqrt{x^2\!-\!a^2}}\;\;\;(> 0).$$ But $\Delta \to 0$ as $x \to \infty$ , whence this branch of the hyperbola approaches unlimitedly from below the line. Accordingly, the line $y = \frac{b}{a}x$ is an asymptote of our curve. By the symmetry, the hyperbola (1) has two asymptotes

(3)

Both asymptotes form with the transversal axis an angle whose tangent is equal to $\frac{b}{a}$ . This equals 1, when the transversal axis and the conjugate axis are equal ($a = b$ ); then the rectangle is a square and one speaks of a rectangular hyperbola. See also the entry transition to skew-angled coordinates.

The tangent line of the hyperbola (1) is $$\frac{x_0x}{a^2}-\frac{y_0y}{b^2} = 1,$$ where $(x_0,\,y_0)$ is the point of tangency on the hyperbola (see tangent of conic section). The tangent halves the angle between the focal radii drawn from $(x_0,\,y_0)$ .

Bibliography

1

L. LINDELÖF: Analyyttisen geometrian oppikirja. Kolmas painos. Suomalaisen Kirjallisuuden Seura, Helsinki (1924).