tailieunhanh - Handbook of mathematics for engineers and scienteists part 21

The circle tangent to one side of a triangle and to the extensions of the other two sides is called an excircle of the triangle. Hach triangle has three excircles. The center of an excircle (an excenter) is the point of concurrency of two external angle bisectors and an interior angle bisector. | 108 Analytic Geometry whose values do not change under parallel translation and rotation of the coordinate axes and the semi-invariant a32 SH 211 4-4-5-3 whose value does not change under rotation of the coordinate axes. The invariant A is called the large discriminant of equation . The invariant 6 is called the small discriminant. Table 1 presents the classification of second-order curves based on invariants. TABLE Classification of second-order curves Type A 0 T 0 Elliptic T 0 Real ellipse for I2 4S or a11 a22 and a12 0 this is a circle Pair of imaginary straight lines intersecting at 5 0 T 0 Imaginary ellipse no real points a real point ellipse degenerating into a point Hyperbolic 5 0 Hyperbola Pair of real intersecting straight lines degenerate hyperbola a 0 Pair of imaginary parallel straight lines Parabolic 5 0 Parabola a 0 Pair of real parallel straight lines 0 Pair of coinciding straight lines a a single straight line . Characteristic equation of second-order curves. The properties of second-order curves can be studied with the use of the characteristic equation I 211 - X 12 J 0 or A2 - IX 6 0. I 221 222 - X The roots A1 and X2 of the characteristic equation are the eigenvalues of the real symmetric matrix aj and as a consequence are real. Obviously the invariants I and 6 of second-order curves are expressed as follows in terms of the roots A1 and A2 of the characteristic equation I Ai A2 S A1A2. . Centers and diameters of second-order curves. A straight line passing through the midpoints of parallel chords of a second-order curve is called a diameter of this curve. A diameter is said to be conjugate to the chords or to the direction of chords which it divides into two parts. The diameter conjugate to chords forms an angle p with the positive direction of the OX-axis and is determined by the equation a 11 x a12y a13 cos p a21 x a22 y a23 sin y 0. . Second-Order Curves 109 All diameters of