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Chapter 5: Coordinate Geometry

    An ellipse is the set of all points in a plane such that the sum of the distances (focal radii) from two given points (foci) is constant. The equation of an ellipse with the sum of focal radii equal to 2a, where a > b and b² = a² - c ², is x² / a² - y² / b² = 1 for foci (-c, 0) and (c, 0). The standard form of the equation of an ellipse with center (h, k) is

(x - h)² / a² + (y - k)² / b² =1

 

A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances (focal radii) from two given points (foci) is constant. The equation of a hyperbola with the difference of focal radii equal to 2a, where b² = c² - a², is

x² / a² – y² / b² = 1

for foci (-c , 0) and (c , 0). The equations of the asymptotes of hyperbola are y = (b/a)x and y = -(b/a)x. The standard form of the equation of a hyperbola with center (h, k) is (x – h)² / a² – (y – k)² / b² = 1. Equations of the form xy = n, n ¹ 0, also define hyperbolas whose asymptotes are the coordinate axes.

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