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[parent] hyperbola (Topic)

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

$\displaystyle \sqrt{(x+c)^2+y^2}-\sqrt{(x-c)^2+y^2} = \pm2a.$
The simplifying of this, via two squarings, yields the equation of the hyperbola
$\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2} = 1.$ (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 $ \vert x\vert$ increases from $ a$ to infinity, $ \vert y\vert$ 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.

\begin{pspicture}(-5.5,-4.5)(5.5,4) \psaxes[Dx=1,Dy=1]{->}(0,0)(-4.5,-3.5)(4.5,3... ...0.5 exp -1.5 div} \rput(0,-4.5){$\mbox{Graph of \,}4x^2-9y^2=9$} \end{pspicture}

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

$\displaystyle y = \frac{b}{a}\sqrt{x^2-a^2}$
of the hyperbola situated in the first quadrant ($ x > a$) and the line
$\displaystyle y = \frac{b}{a}x.$
The difference of their ordinates corresponding a same abscissa $ x$ may be written
$\displaystyle \Delta = \frac{b}{a}(x-\sqrt{x^2-a^2}) = \frac{ab}{x+\sqrt{x^2-a^2}}\;\;\;(> 0).$
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.

\begin{pspicture}(-5.5,-4.5)(5.5,4) \psaxes[Dx=10,Dy=10]{->}(0,0)(-4.5,-3.5)(4.5... ...c{y^2}{b^2}=1$\mbox{\, with asymptotes \,}$y = \pm\frac{b}{a}x$} \end{pspicture}

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

$\displaystyle \frac{x_0x}{a^2}-\frac{y_0y}{b^2} = 1,$
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).



"hyperbola" is owned by pahio. [ full author list (2) ]
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See Also: conic section, Euclidean distance, transition to skew-angled coordinates, tangent line, hyperbolic rotation, graph of equation $\,xy =$ constant

Also defines:  hyperbola, focus, foci, focal radius, focal radii, apex of hyperbola, apices of hyperbola, transversal axis, conjugate axis, rectangular hyperbola

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unit hyperbola (Definition) by pahio
conjugate hyperbola (Definition) by pahio
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Cross-references: tangent of conic section, tangent line, transition to skew-angled coordinates, tangent, angle, sides, rectangle, diagonals, symmetry, curve, asymptote, abscissa, ordinates, difference, quadrant, transversal, Pythagorean theorem, line, infinity, real, clear, symmetric, squares, origin, axis, coordinate, equation, line segments, distances, Euclidean plane, points, locus
There are 28 references to this entry.

This is version 27 of hyperbola, born on 2007-05-21, modified 2008-05-31.
Object id is 9427, canonical name is Hyperbola2.
Accessed 6061 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
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