Define Hyperbola
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.
Eccentricity of Hyperbola
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.
Standard Equation of Hyperbola
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
Solved Examples
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
Did you know?
The design of cooling towers in Nuclear reactors are hyperbolic structures. They help make the structure durable, efficient, and cost-effective.
FAQs on Hyperbola
1. What is the fundamental difference between a hyperbola and a parabola?
The main difference lies in their definitions and the number of focal points. A hyperbola is the set of all points where the difference of the distances from two fixed points (foci) is constant. In contrast, a parabola is the set of all points that are equidistant from a single fixed point (focus) and a single fixed line (directrix). Other key differences include:
- A hyperbola has two branches and two asymptotes, while a parabola has a single branch and no asymptotes.
- The eccentricity (e) of any hyperbola is always greater than 1 (e > 1), whereas for a parabola, it is always equal to 1 (e = 1).
2. What is the standard equation for a hyperbola as per the Class 11 syllabus?
For the CBSE Class 11 curriculum, a hyperbola centred at the origin has two standard equations, depending on its orientation:
- Horizontal Hyperbola: The equation is x²⁄a² - y²⁄b² = 1. Here, the transverse axis is along the x-axis.
- Vertical Hyperbola: The equation is y²⁄a² - x²⁄b² = 1. Here, the transverse axis is along the y-axis.
3. What are the two main types of hyperbolas based on their axis alignment?
The two main types are distinguished by which axis contains the foci and vertices:
- Transverse Axis along the X-axis (Horizontal Hyperbola): This type opens left and right. Its vertices are at (±a, 0) and foci are at (±c, 0). Its standard equation is x²⁄a² - y²⁄b² = 1.
- Transverse Axis along the Y-axis (Vertical Hyperbola): This type opens up and down. Its vertices are at (0, ±a) and foci are at (0, ±c). Its standard equation is y²⁄a² - x²⁄b² = 1.
4. How do you calculate the length of the latus rectum for a hyperbola?
The latus rectum of a hyperbola is a line segment perpendicular to the transverse axis, passing through a focus, with its endpoints on the curve. Its length is a key property that helps define the 'width' of the hyperbola at its focus. For a standard hyperbola with the equation x²⁄a² - y²⁄b² = 1 or y²⁄a² - x²⁄b² = 1, the length of the latus rectum is calculated using the formula 2b²/a.
5. What are some real-world examples of hyperbolas?
Hyperbolic shapes appear in various real-world scenarios and technologies. Some common examples include:
- The shape of cooling towers at nuclear power plants, which offers structural stability and efficient cooling.
- The path of a spacecraft or comet that swings by a planet but has enough velocity to escape its gravity.
- The design of hyperbolic mirrors and lenses used in advanced telescopes.
- In architecture, some modern building designs use hyperbolic paraboloid roofs for their strength and aesthetic appeal.
6. How is a hyperbola defined as a conic section?
A hyperbola is formed when a flat plane intersects a double-napped cone. Specifically, for a hyperbola to be created, the plane must cut through both nappes (the top and bottom halves) of the cone. This happens when the angle of the intersecting plane, relative to the cone's central axis, is smaller than the angle of the cone's generator line. This unique intersection results in the two distinct, open curves that define the hyperbola.
7. What does the eccentricity of a hyperbola tell us about its shape?
The eccentricity (e) of a hyperbola is a measure of how much it deviates from being a parabola or an ellipse. For any hyperbola, the eccentricity is always greater than 1 (e > 1). The value of 'e' directly influences the shape of the hyperbola's branches: a value of 'e' closer to 1 results in sharper, more pointed curves, while a larger value of 'e' produces flatter, more open curves.
8. Does the right-hand side of a hyperbola's equation always have to equal 1?
No, not necessarily. The form where the equation equals 1, such as x²⁄a² - y²⁄b² = 1, is known as the standard form. This convention makes it easy to identify the vertices and calculate other properties. However, a hyperbola can be represented by a general equation like Ax² - By² + Cx + Dy + E = 0, which can be rearranged and simplified into the standard form. The '1' is a result of standardization, not a universal requirement for all forms of the equation.
9. What defines a rectangular hyperbola and why is it considered special?
A rectangular hyperbola is a special type of hyperbola where its asymptotes are perpendicular to each other. This occurs when the lengths of the semi-transverse axis (a) and the semi-conjugate axis (b) are equal (a = b). Its eccentricity is always fixed at √2. The standard equation x²⁄a² - y²⁄a² = 1 simplifies to x² - y² = a². It is special because when rotated by 45°, its equation simplifies to the very useful form xy = c², which is fundamental in representing inverse relationships in physics and economics.
