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The Hyperbola

The Hyperbola. A Look into the Polar Aspect. Degenerate Case. Definition: A limiting case in which a class of object changes its nature to belong to another simpler class. The Hyperbola Degenerate: Where two lines cross through the Pole and intersect. Application.

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The Hyperbola

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  1. The Hyperbola A Look into the Polar Aspect

  2. Degenerate Case • Definition: A limiting case in which a class of object changes its nature to belong to another simpler class. • The Hyperbola Degenerate: Where two lines cross through the Pole and intersect.

  3. Application • In radar tracking stations, an object is • identified by the system shooting out • signals from two point sources. • The concentric circles of these sound • waves intersect in hyperbolas, providing • an application for the use of • hyperbolas..

  4. The standard form of the hyperbola • The Standard Form Equation of a Hyperbola is: • 1. (x-h)² - (y-k)² = 1 • a² b² • (y-k)² - (x-h)² = 1 • b² a² • The Geometric/ Algebraic Form: • y=a/(x-h) + k

  5. The key words/relationships: Hyperbolas • The major terms are: • Asymptotes- The diagonal lines that restrict the graph • Axis- Transverse and Conjugate • Branches- The 2 curved lines in the graph • Center- The midpoint between the 2 branches • Foci- A point that is equidistant from the branches • 1, x² - y² = 1 b²=c²-a² • a² b² • 2. y² - x² = 1 b² = c² - a² • a² b² Center- (0,0) Foci- (Transverse Axis along X-Axis

  6. Rotated form/ Conic Form • All conics take the form of: • Ax²+Bxy+Cy²+Dx+Ey+F=0 • If B²-4AC > 0; the graph is a hyperbola • Rotated Hyperbola if the Bxy term is present in the equation.

  7. Eccentricity • Definition- how much a conic section varies from being circular • The eccentricity of a hyperbola is it must equal greater than one • e= a²+b² • a

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