23 junio, 2024

Hyperbola: characteristics, formulas, elements, examples

What is a hyperbola?

The hyperbola is the set of points in the plane such that the absolute value of the difference between the distances to two fixed points, called foci, remains constant. This set of points forms the curve with two branches that is observed in figure 1.

There is shown a point P(x,y), the foci F1 and F2 separated by a distance equal to 2c. The mathematical way of expressing this relationship is through:

All the points of the hyperbola satisfy this condition, which leads to the equation of the hyperbola, as will be seen later. The midpoint between the foci is called the center C and in the figure it coincides with the point (0,0), but the hyperbola can also be displaced and its center corresponds to another point with coordinates C (h,k).

In the upper figure, the x-axis is the focal axis of the hyperbola, since the foci are there, but one can also be built whose focal axis is the y-axis.

The hyperbola is part of the curves known as conical, which are so named because they can be derived from cutting a cone with a flat section. A hyperbola is obtained by intersecting the cone and the plane, provided that the plane does not pass through the vertex of the cone and the angle formed by the plane with the axis of the cone is less than the angle it forms with its generatrix axis.

Along with the parabola, the circle and the ellipse, the conics have been known since ancient times. The Greek mathematician Apollonius of Perga (262-190 BC) wrote a geometry treatise detailing their properties and he himself gave them the names by which they are known to this day.

Characteristics of the hyperbola

These are some of the most outstanding characteristics of a hyperbola:

It is a flat curve, therefore it is enough to give the coordinates (x,y) of each point that belongs to it.
It is also an open curve, unlike a circle or an ellipse.
It has two symmetrically arranged branches.
Both the vertical axis and the horizontal axis can be considered as axes of symmetry, but the axis where the foci are located is called focal axis or principal axis.
It is symmetric about its center.
The hyperbola intersects the focal axis at two points called verticesthat is why the focal axis is sometimes called royal axiswhile the other axis is called imaginary axisbecause it has no points in common with the hyperbola.
The center of the hyperbola is located halfway between the points called the foci.
It is associated with two very particular lines called asymptotes, which are lines that the hyperbola approaches, but does not cross, when the values ​​of x and y are very large. The asymptotes intersect at the center of the hyperbola.

equations and formulas

Equation of the hyperbola with center at (0,0)

Starting from the definition given at the beginning:

This positive constant is usually called 2a and it is the distance that separates the vertices of the hyperbola, so:

On the other hand, dP1, dP2, and 2c are the sides of the triangle shown in Figure 1, and by elementary geometry, the subtraction of the squares of the sides of any triangle is always less than the square of the remaining side. So:

4a2 < 4c2

AND:

a < c

This result will be useful shortly.

Since the distance between two points P1(x1,y1) and P2(x2,y2) is:

Substituting the coordinates P(x,y), F1(-c,0) and F2(c,0) leaves:

Which is equivalent to:

Squaring both members to eliminate the roots and rearranging the terms, we arrive at:

The quantity c2 – a2, which is always a positive quantity because a < c, is called b2, so the above can be rewritten as:

b2x2 – a2y2 = a2 b2

Dividing all the terms by a2 b2, we get the equation of the hyperbola centered at (0,0) with the real axis horizontal:

With a and b greater than 0. This equation is called canonical equation of the hyperbola and the denominator a2 always corresponds to the positive fraction.

The hyperbola centered at (0,0) and with the real axis vertical takes the form:

Intersections of the hyperbola with the coordinate axes

The intersections of the hyperbola with the coordinate axes are found by setting y = 0 and x = 0 respectively in the equation:

For y = 0

x2 /a2 = 1 ⇒ x2 = a2

x = ± a

The hyperbola intersects the x-axis at two points called vertices, whose respective x-coordinates are: x = a and x = -a

For x = 0

One obtains -y2 /b2 = 1, which has no real solution and it follows that the hyperbola does not intersect the vertical axis.

Equation of the hyperbola with center at (h,k)

If the center of the hyperbola is at the point C(h,k), then its canonical equation is:

Elements of the hyperbola

Center

It is the midpoint of the segment F1F2 and its coordinates are (h,k) or (xo,yo).

spotlights

They are the two fixed points F1 and F2 that are on the real axis of the hyperbola, with respect to which the difference of distances to the point P(x,y) remains constant. The distance between the foci and the center of the hyperbola is “c”.

vector radius

This is the name given to the distance between a point P and one of the foci.

Focal distance

It is the distance that separates both foci and is equal to 2c.

vertices

The vertices V1 and V2 are the points where the hyperbola intersects the real axis. A vertex and the center of the hyperbola are separated by the distance a, therefore the distance between the vertices is 2a.

focal axis, principal axis or real axis

It is the axis where the foci are located and measures 2c. It can be located on any of the two Cartesian axes and the hyperbola intersects it at points called vertices.

Transverse axis, secondary axis or imaginary axis

It is the axis perpendicular to the focal axis and measures 2b. The hyperbola does not intersect it, which is why it is also called the imaginary axis.

asymptotes

They are two straight lines, whose respective slopes are m1 = (b/a) and m2 = − (b/a), which intersect at the center of the hyperbola. The curve never intersects these lines and the product between the distances from any point on the hyperbola to the asymptotes is constant.

To find the equations of the asymptotes, it is enough to set the left-hand side of the canonical equation of the hyperbola equal to 0. For example, for the hyperbola centered at the origin:

hyperbola rectangle

It is the rectangle whose width is the distance between the vertices 2a and the distance 2b and is centered in the center of the hyperbola. Its construction facilitates the manual drawing of the hyperbola.

straight side

String passing through one of the foci, perpendicular to the real axis.

Eccentricity

It is defined as the quotient between the focal length and the real axis:

e = each

It is always greater than 1, since c is greater than a, and less than √2.

The value of e indicates whether the hyperbola is rather closed (narrow rectangle, elongated towards the principal axis) or open (wide rectangle, elongated towards the imaginary axis).

Tangent line to the hyperbola at the point P(x1,y1)

A tangent line to the hyperbola at a point P(x1,y1) of the hyperbola is the bisector of the two radii vectors of said point.

For a hyperbola with the principal axis parallel to the x-axis, the slope of the tangent line to the hyperbola at a point P(x1,y1) is given by:

And if the hyperbola has a principal axis parallel to the y axis, then:

Examples of hyperbola

Scattering of alpha particles by a nucleus

When bombarding atomic nuclei with alpha particles, which are nothing more than helium nuclei, these are repelled, since any atomic nucleus has a positive charge. These helium nuclei are scattered following hyperbolic trajectories.

Trajectories of the bodies of the solar system

In the solar system, objects move under the action of the force of gravity. The description of the motion is derived from a differential equation in which the force is conservative and inversely proportional to the square of the distance. And the solutions of said equation are the possible trajectories that the objects follow.

Well, these trajectories are always conical: circles, ellipses, parabolas or hyperbolas. The first two are closed curves, and that’s how the planets move, but some comets follow open paths, like parabolas or hyperbolas, with the Sun at one focus.

sound minima

When you have two sound sources, such as two loudspeakers that emit sounds uniformly in all directions, located along a straight line, the sound intensity minima (destructive interference) lie on a hyperbola whose main axis is that line, and at the foci of the hyperbola are the loudspeakers.

solved exercise

Find the elements of the following hyperbola: vertices, foci and asymptotes of the hyperbola and construct its graph:

Solution

The center of this hyperbola coincides with the origin of the coordinates and its real axis is horizontal, since the positive fraction corresponds to the variable x.

The semiaxes of the hyperbola are:

a2 = 16 ⇒ a = 4

b2 = 4 ⇒ b = 2

Thus, the center rectangle is 4 units wide and 2 units tall. Remembering that above it was mentioned that c2 – a2 = b2 , then:

c2 = a2 + b2 ⇒ c2 = 16 + 4 = 20

Therefore, the semi-focal length is:

c = √20 = 2√5

And the foci are at the coordinate points F1 (-2√5,0) and F2 (2√5,0).

The slopes of the asymptotes are:

m = ±(b/a) = ±(2/4)=±0.5

Therefore the respective equations of each one are:

y1 = 0.5x ; y2 = -0.5x

The hyperbola can be easily graphed through online software such as Geogebra:

References

Physicalab. Equation of the hyperbola. Retrieved from: physicalab.com
Hoffman, J. Selection of Mathematics topics. Volume 2.
Stewart, J. 2006. Precalculus: Mathematics for Calculus. 5th. Edition. Cengage Learning.
Universe Formulas. the hyperbola. Retrieved from: universoformulas.com
Zill, D. 1984. Algebra and Trigonometry. McGraw Hill.

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