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# hyperbola

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 ($\pm 2a$). 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

$\sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}=\pm 2a.$ |

The simplifying of this, via two squarings, yields the equation of the hyperbola

$\displaystyle\frac{x^{2}}{a^{2}}\!-\!\frac{y^{2}}{b^{2}}\;=\;1.$ | (1) |

Here we have denoted $c^{2}\!-\!a^{2}=b^{2}$ where $b>0$.

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

$\displaystyle y\;=\;\pm\frac{b}{a}\sqrt{x^{2}\!-\!a^{2}}.$ | (2) |

This shows that $y$ is real only for $x\geqq a$; for $x=\pm a$ we have $y=0.$ When $|x|$ increases from $a$ to infinity, $|y|$ increases from $0$ to infinity. So we see that the hyperbola consists of two distinct branches from which the one is to the right from the line $x=a$ and the other to the left from the line $x=-a$. These lines touch the branches at the points $(a,\,0)$ and $(-a,\,0)$, which are called the apices of the hyperbola.

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

$\displaystyle y\;=\;\pm\frac{b}{a}x.$ | (3) |

The asymptotes are easy to draw, since they are the lengthened diagonals of the rectangle whose sides are on the lines $x=\pm a$ and $y=\pm b$. The hyperbola may be sketched by utilising that rectangle and the asymptotes.

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.

# References

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

## Mathematics Subject Classification

53A04*no label found*51N20

*no label found*51-00

*no label found*

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