tailieunhanh - Handbook of mathematics for engineers and scienteists part 20

The circle tangent to the three sides of a triangle and lying inside the triangle is called the incircle of the triangle. The center O-y of the incircle (the incemer) is the point where the angle bisectors meet (Fig. ). The straight lines connecting the vertices of a triangle with the points at which the incircle is tangent to the respective opposite sides intersect in a single point G called the Gergonne point (Fig. ). | . Second-Order Curves 101 . Ellipse in polar coordinate system. In polar coordinates p p the equation of an ellipse becomes P P I-------- 1 - e cos p where 0 p 2n. . Hyperbola . Definition and canonical equation of hyperbola. A curve on the plane is called a hyperbola if there exists a rectangular Cartesian coordinate system OXY in which the equation of this curve has the form x2 a2 y- i b2 1 where a 0 and b 0 see Fig. . The coordinates in which the equation of a hyperbola has the form are called the canonical coordinates for the hyperbola and equation itself is called the canonical equation of the hyperbola. The hyperbola is a central curve of second order. It is described by equation and consists of two connected parts arms lying in the domains x a and x -a. The hyperbola has two asymptotes given by the equations y x and y - x. a a More precisely its arms lie in the two vertical angles formed by the asymptotes and are called the left and right arms of the hyperbola. A hyperbola is symmetric about the axes OX and OY which are called the principal real or focal and imaginary axes. The angle between the asymptotes of a hyperbola is determined by the equation p b tan 2a and if a b then p 2n an equilateral hyperbola . 102 Analytic Geometry The number a is called the real semiaxis and the number b is called the imaginary semiaxis. The number c Va2 b2 is called the linear eccentricity and 2c is called the focal distance. The number e c a Va2 b2 a where obviously e 1 is called the eccentricity or the numerical eccentricity. The number p b2 a is called the focal parameter or simply the parameter of the hyperbola. The point 0 0 0 is called the center of the hyperbola. The points A1 -a 0 and Az a 0 of intersection of the hyperbola with the real axis are called the vertices of the hyperbola. Points F1 -c 0 and F2 c 0 are called the foci of the hyperbola. This is why the real axis of a .