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Parametric equations are crucial for understanding hyperbolas, a type of conic section. These equations, using a parameter t, allow for detailed analysis of hyperbolas' geometric properties. They are essential in fields like astronomy and engineering, where they model the motion of celestial bodies and the design of optical systems. The text delves into deriving these equations from a hyperbola's standard form and their real-world applications.
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Parametric equations are a versatile method for representing the coordinates of points on various curves, including the hyperbola
Use in Astronomy, Physics, and Engineering
Parametric equations are invaluable in disciplines such as astronomy, physics, and engineering, where understanding the motion and relationships of objects in space is crucial
Use in Design and Analysis
Parametric equations are useful in the design and analysis of various technological and scientific endeavors, such as hyperbolic mirrors and celestial mechanics
Parametric equations allow for a more comprehensive exploration of the properties of hyperbolas compared to Cartesian equations
The standard form of a hyperbola includes a center, transverse and conjugate axes, and distances to the vertices and co-vertices
Horizontal hyperbolas have a standard form of \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) and parametric equations of \( x = h + a \cdot \cosh(t) \) and \( y = k + b \cdot \sinh(t) \)
Vertical hyperbolas have a standard form of \(\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1\) and parametric equations of \( x = h + a \cdot \sinh(t) \) and \( y = k + b \cdot \cosh(t) \)
Parametric equations for hyperbolas are derived by substituting specific values into general parametric forms based on the orientation of the hyperbola's transverse axis
The accuracy of parametric equations can be verified by eliminating the parameter and showing that the resulting equation conforms to the hyperbola's standard form
Hyperbolic trigonometric identities, such as \( \cosh^2(t) - \sinh^2(t) = 1 \), are used to demonstrate the validity of parametric equations for hyperbolas
Ellipses can be represented using the parametric equations \( x = h + a \cdot \cos(t) \) and \( y = k + b \cdot \sin(t) \)
Parabolas can be expressed parametrically using a single parameter \( t \) and a set of equations involving the focus and directrix
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Standard form of horizontal hyperbola
Given by (x-h)^2/a^2 - (y-k)^2/b^2 = 1; (h, k) is center, a is distance to vertices, b to co-vertices.
01
Standard form of vertical hyperbola
Given by (y-k)^2/b^2 - (x-h)^2/a^2 = 1; (h, k) is center, b is distance to vertices, a to co-vertices.
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Difference between transverse and conjugate axis
Transverse axis is line through vertices; conjugate axis is line through co-vertices, perpendicular to transverse.
