Hyperbola Notes PDF

# अतिपरवलय (Hyperbola) अतिपरवलय एक ऐसे बिन्दु का बिन्दुपथ है जिसकी स्थिर बिन्दु से दूरी अधिक व स्थिर रेखा से दूरी कम होती है। - स्थिर बिन्दु = नाभि - स्थिर रेखा = नियता <br> ## Diagram: The image shows a diagram of a hyperbola with the following elements: - **Foci:** A and A' - **Vertices:** A...

# अतिपरवलय (Hyperbola) अतिपरवलय एक ऐसे बिन्दु का बिन्दुपथ है जिसकी स्थिर बिन्दु से दूरी अधिक व स्थिर रेखा से दूरी कम होती है। - स्थिर बिन्दु = नाभि - स्थिर रेखा = नियता <br> ## Diagram: The image shows a diagram of a hyperbola with the following elements: - **Foci:** A and A' - **Vertices:** A and A' - **Center:** O - **Directrix:** The vertical lines on either side of the hyperbola - **Transverse axis:** The horizontal axis AA' - **Conjugate axis:** The vertical axis that intersects the hyperbola at the center - **Asymptotes:** The lines y= b/a x and x=a/b *x* <br> ## Formulas: The equation of a hyperbola is: $x^2/a^2 - y^2/b^2 = 1$ where: - a is the distance from the center to each vertex. - b is the distance from the center to each end of the conjugate axis. The eccentricity of a hyperbola is: $e = \sqrt{1+b^2/a^2}$ The focal length of a hyperbola is: $ae = \sqrt{a^2 + b^2}$ The equation of the directrix of a hyperbola is: $x = \pm a/e$ The equation of the latus rectum of a hyperbola is: $x = \pm ae$ <br> ## Properties: - The hyperbola is symmetric about both the x-axis and the y-axis. - The distance between the two foci is 2ae. - The distance between the two vertices is 2a. - The distance between the two directrices is 2a/e. - The length of the latus rectum is 2b²/a. - The asymptotes of a hyperbola are the lines y= b/a x and x=a/b *x*. - The hyperbola has two branches, which open up and down or left and right depending on the orientation of the transverse axis. <br> ## Related Terms: - **Focus:** A point that is used to define the hyperbola. - **Vertex:** A point where the hyperbola intersects the transverse axis. - **Directrix:** A line that is used to define the hyperbola. - **Eccentricity:** A measure of the shape of the hyperbola. - **Latus rectum:** A line segment that passes through a focus and is perpendicular to the transverse axis. - **Asymptote:** A line that the hyperbola approaches as it extends to infinity. - **Conjugate axis:** The axis that is perpendicular to the transverse axis. - **Transverse axis:** The axis that intersects the hyperbola at the vertices. <br> # Types of Hyperbolas There are two main types of hyperbolas: ## 1. Rectangular Hyperbola Also known as an equilateral hyperbola, this type has an equal length for both its transverse and conjugate axes: - The equation for a rectangular hyperbola is: $x^2 - y^2 = a^2$ - The eccentricity of a rectangular hyperbola is: $e = \sqrt{2}$ ## 2. Conjugate Hyperbola This type of hyperbola is formed by switching the transverse and conjugate axes of a main hyperbola. - The equation of a conjugate hyperbola is: $-x^2/a^2 + y^2/b^2 = 1$ # Important Points - The focal length of a hyperbola is always greater than the distance from the center to a vertex. - The eccentricity of a hyperbola is always greater than 1. - The asymptotes of a hyperbola are lines that pass through the center of the hyperbola and are perpendicular to the transverse axis. # Applications - Hyperbolas are used in many areas of physics, engineering, and astronomy. - They are used to model the trajectories of comets and other celestial bodies. - They are used to design antennas, reflectors, and lenses. - They are used to track the path of a sound wave or light wave. # Summary A hyperbola is a conic section with two branches that open up and down or left and right depending on the orientation of the transverse axis. It is defined by the property that every point on the hyperbola has a constant difference in distance from the two foci and the directrices. Hyperbolas are used in many areas of physics, engineering, and astronomy.