According to the Hyperbola definition, it is a collection of points in the plane such that there is a constant distance between two fixed points and each point. Hyperbola is made up of two similar curves that resemble a parabola.

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Let us consider the Hyperbola in the above diagram. The Two fixed points \[F_{1}\] and \[F_{2}\] in the diagram are known as foci or focus. Now consider three points on the hyperbola \[P_{1}\], \[P_{2}\], and \[P_{3}\]. According to the definition of a hyperbola, we determine that \[P_{1}\]\[F_{1}\] + \[P_{1}\]\[F_{2}\] is equal to constant. Similarly, \[P_{2}\]\[F_{1}\] + \[P_{2}\]\[F_{2}\] and \[P_{3}\]\[F_{1}\]+\[P_{3}\]\[F_{2}\] are also constant.

\[P_{1}\]\[F_{1}\]+\[P_{1}\]\[F_{2}\]=\[P_{2}\]\[F_{1}\]+\[P_{2}\]\[F_{2}\]=\[P_{3}\]\[F_{1}\]+\[P_{3}\]\[F_{2}\]= Constant

So, when we join both the foci using a line segment, then its midpoint gives us centre (O). Hence, this line segment is known as the transverse axis.

The eccentricity of Hyperbola is the distance ratio from the centre to a vertex and from the centre to a focus(foci). The eccentricity of Hyperbola formula can be showed as follows:

Eccentricity(e)=\[\frac{c}{a}\]

Eccentricity is never less than 1 for Hyperbola. Since c ≥ a.

Let us now derive the standard equation of Hyperbola. For this, consider a hyperbola with centre (O) at (0,0) and its foci lie on any one of the x or y-axis. Look at the diagram below to understand the concept thoroughly.

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Both the foci's lie at a distance of "c" on the x-axis and the vertices are at a distance "a" from (0,0) origin. Let us consider a point Z on the Hyperbola so that it satisfies the definition Z\[F_{1}\]+Z\[F_{2}\] is constant 2a.

2a = Z\[F_{1}\]+Z\[F_{2}\]

According to distance formulae

\[\sqrt{(x + c)^{2} + y^{2}}\] - \[\sqrt{(x - c)^{2} + y^{2}}\] = 2a --equation(1)

\[\sqrt{(x + c)^{2} + y^{2}}\] = 2a + \[\sqrt{(x - c)^{2} + y^{2}}\]

Squaring both sides

\[\frac{x^{2}}{a^{2}}\] - \[\frac{y^{2}}{c^{2}}\] - \[a^{2}\] = 1

Since we know that \[b^{2}\] = \[c^{2}\] - \[a^{2}\] and 0 < a < c.

\[y^{2}\] = \[b^{2}\]\[(\frac{x^{2}}{a^{2}} - 1)\]

Substituting \[y^{2}\] in equation(1)

\[\frac{x^{2}}{a^{2}}\] - \[\frac{y^{2}}{b^{2}}\] = 1

Example 1:

Given the hyperbola \[\frac{x^{2}}{9}\] - \[\frac{y^{2}}{25}\] = 1 Then, what is the position of Point P(6,-5)?

Answer:

Hyperbola definition fits perfectly for \[\frac{x^{2}}{9}\] - \[\frac{y^{2}}{25}\] = 1

According to the standard equation of Hyperbola

\[\frac{x^{2}}{a^{2}}\] - \[\frac{y^{2}}{b^{2}}\] = 1

Rearranging this equation, we know that P lies either inside or outside of the Hyperbola.

Hence,

\[\frac{x^{2}}{a^{2}}\] - \[\frac{y^{2}}{b^{2}}\] - 1 < 0, or > 0

Substitute the values of x, y, a, and b in this equation.

\[\frac{6^{2}}{9}\] - \[\frac{(-5)^{2}}{25}\] - 1

Further solving this equation

= \[\frac{36}{9}\] - \[\frac{25}{25}\] - 1

= 4 - 1 - 1

= 2

Since 2>0, we can conclude that the point P lies inside the given Hyperbola.

Example 2:

Find the equation of Hyperbola whose vertices are (9,2) and (1,2) as well as the distance between the foci is 10.

Answer:

According to the meaning of Hyperbola the distance between foci of Hyperbola is 2ae

2ae=10

In the eccentricity of Hyperbola formula

ae=5 --(1)

Since both, the vertices are at two on the y-axis.

We can calculate the centre of the Hyperbola by finding the midpoint of vertices.

\[\frac{(9+1)}{2}\], \[\frac{(2+2)}{2}\]

=(5,1)

Assume the equation of this hyperbola is \[\frac{(x-p)^{2}}{a^{2}}\] - \[\frac{(y-q)^{2}}{b^{2}}\] = 1

length of the transverse axis= 8

2a=8

a=4

Using (1)

4e=5

e=5/4

We know that \[b^{2}\] = \[a^{2}\](\[e^{2}\] - 1)

Substituting respective values

\[b^{2}\] = 9

Now let’s assemble the equation

\[\frac{(x-p)^{2}}{a^{2}}\] - \[\frac{(y-q)^{2}}{b^{2}}\] = 1

\[\frac{(x-5)^{2}}{16}\] - \[\frac{(y-2)^{2}}{9}\] = 1

Solving the equation further

\[9x^{2} - 16y^{2} - 90x - 64y + 17\] = 0

The design of cooling towers in Nuclear reactors are hyperbolic structures. They help make the structure durable, efficient, and cost-effective.

FAQ (Frequently Asked Questions)

1. Differentiate Between Hyperbola and Parabola?

When we observe the meaning of Hyperbola, it seems precisely like two parabola's opposite to each other. But that is not the case despite having the same curve structure; these two are very different from each other. In Hyperbola, we calculate how distant a set of points are from two fixed points whereas in the parabola a set of points is at equal distances from the directrix. One major difference between them is that all parabolas have the same shape whereas all hyperbolas have different shapes. You may also notice that the two arms in a parabola are parallel, but that is not the case for hyperbolas.

2. What is the Length of the Latus Rectum in the Case of a Hyperbola?

Latus rectum is a chord parallel to the directrix and passing through focus. You can observe that a hyperbola has two latus rectums. In the below diagram the latus rectum would be CD or AB. We can calculate the length of the latus rectum using the standard equation.

b²/a is the length of the semi latus rectum, and 2(b²/a) gives us the length of the latus rectum. We can also calculate the latus rectum using another formulae 2a(e² - 1). We use a latus rectum in the field of astronomy to study planetary motion in outer space.