Propositions of a Hyperbola

Geometrical mathematics revolves around the surface area and volume definitions. The concept of propositions of hyperbola forms an integral part of mathematics as well as physics. The concept tries to derive relationships between various variables that are often used in physics and maths. This article explores the propositions of a hyperbola meaning and propositions of hyperbola examples to create a deep understanding of the most commonly taught concept that is a part of the everyday life of individuals.

Surface Area and Volume

The definitions of surface area and volume set the basics for different propositions of a hyperbola. The concept tries to define several propositions of a hyperbola and derives various solid conclusions based on the same, which can be applied in daily life. The surface area and volume definitions are given below.

Surface Area

The entire area taken up by the surface of an object is said to be its surface area. Varying 3D shapes have different surface areas in geometry, which may readily be determined using the formulas.

Volume

Volume is a three-dimensional space bounded by a boundary or occupied by an object in mathematics.

Propositions of a Hyperbola

A hyperbola is a conic with a larger eccentricity than unity. It is the locus of a point whose distance from a fixed point called focus and from a fixed-line called directrix is a constant called ‘e,’ which, in this case, is more than one.  The examples and the propositions of hyperbola meaning are given below.

Diameter of a Hyperbola

A hyperbola’s diameter refers to the location of the centre point of a system of parallel chords. The equation of the diameter for a hyperbola is y = b2x/(a2m), where m is the slope of the system of parallel chords.

Conjugate Diameters

Conjugate diameters are the two circumferences of a hyperbola that intersect chords parallel to each other. Therefore, the diameters y = mx and y = m1x of the hyperbola x²/a² – y²/b² = 1 are conjugate if mm1 = b²/a².

Latus Rectum

The latus rectum is the chord of a hyperbola that passes through one of the foci and is perpendicular to the transverse axis.

∴ If the latus rectum’s length is 2l, the coordinates of one of its extremities are (ae, l).

∴ We have (ae, l) because it is on the hyperbola x²/a2 – y2/b2.

1 = e² (l²/b²)

Since b² = a²(e² – 1),

l² = b²(e²–1)

= b²a²(e²–1)/a²

= b⁴/a²

l = b²/a

Since the latus rectum length is 2I, double the obtained value of I.

∴ 2b²/a is the latus rectum’s length.

Director Circle of a Hyperbola

The point of intersection of two perpendicular tangents to a hyperbola is located on the director circle.

The hyperbole’s director circle, x2/a2 – y2/b2 = 1, has the equation x2 + y2 = a2 – b2, i.e., a circle whose centre is origin and radius is (a2 – b2).

Note that:

• This circle is real but with radius greater than 0 if b2 is less than a2.
• When b² = a², the circle’s radius is 0, and it becomes a point circle at the origin. In this situation, the centre is the only place from which tangents to the hyperbola may be formed at a right angle.
• If b² > a², the circle’s radius is imaginary; hence there is no such circle, and no right-angle tangents can be made to the circles.

Examples of Hyperbola

Now, let’s discuss propositions of hyperbola examples. 

Tangents are drawn to the circle x² + y² = 9 from any point on the hyperbola x2/9 – y2/4 = 1. Locate the chord of contact’s midpoint locus.

Let any point on the hyperbola be (3 sec θ, 2 tan θ).

Therefore, the chord of contact of the circle x² + y² = 9 concerning the point (3 sec θ, 2 tan θ) is,

(3 sec θ)x + (2 tan θ)y = 9.  …… (1)

Let (x1, y1) be the midpoint of the chord of contact.

Then the equation of the chord in midpoint form is

xx1 + yy1 = x12 + y12 …. (2)

As both the above equations are equal, so

3 sec θ/x1 = 2 tan θ/y1 = 9 (x12 + y12)

This gives sec θ = 9×1/3(x12 + y12)

and tan θ = 9y1/2(x12 + y12)

On eliminating θ from the above two equations, we get

81×12/ 9(x12 + y12)2 – 81y12/4(x12 + y12)2 = 1.

Therefore, the required locus is x2/9 – y2/4 = (x2 + y2)2/ 81.

Conclusion

It could be concluded that there are various propositions of a hyperbola meaning, these are widely applied in daily life. These propositions form the basics of the fields of mathematics and science and are frequently used concepts. They set relationships between variables such as focal distances to determine the basic shape of a lens worn by myopic or hyper myopic people. Thus, surface area and volume definitions are directly or indirectly a vital part of routine life. The propositions of hyperbola examples are also provided for the convenience of understanding.