where h is the length of the altitude drawn to the hypotenuse; moreover, the altitude cuts the hypotenuse into segments of lengths m and n. In a right triangle, the length of the median mc drawn from the vertex of the right angle coincides with the circumradius B and is equal to half the length of the hypotenuse c7 mc = R = jc. The inradius is given by the formula r = -(a -\-b-c). The area of the right triangle is S = aha = -a5(see also Paragraphs to ). | 150 Analytic Geometry TABLE Classification of quadrics central surfaces 5 0 Class 0 or 7 0 SS 0 and 7 0 but not both Nondegenerate surfaces A 0 A 0 Ellipsoid x2 y2 z2 a2 b2 c2 Two-sheeted hyperboloid x2 y2-z2 _ -t a2 b2 c2 A 0 Imaginary ellipsoid x 4 - a2 b2 c2 One-sheeted hyperboloid x2 f- f _ 1 a2 b2 c2 Imaginary cone with real vertex Real cone Degenerate surfaces A 0 S -1 o XL y zL _ 0 a2 b2 c2 TABLE Classification of quadrics central surfaces S 0 Class ar Nondegenerate surfaces A 0 Degenerate surfaces A 0 id type A 0 7 0 7 0 Hyperbolic paraboloid x2 y2 _ 7 7 21 7 0 A 0 Cylindrical surfaces a 0 Elliptic paraboloid x2 y2 p f 2Z Elliptic cylinder Hyperbolic cylinder .2 2 x y _ 1 a2 b2 Parabolic cylinder y2 _ 2px Imaginary aS 0 x2 y2 _ -i a2 b2 1 Real aS 0 x2 i a2 b2 1 Reducible surfaces a 0 Pair of imaginary planes intersecting in a real straight line x2 y2 77 _ 0 a2 b2 Pair of real intersecting planes 2 2 x y _ 0 a2 b2 0 Pair of real reducible planes S _ 0 x2 _ 0 Pair of imaginary parallel planes S 0 2 2 x _ -a Pair of real parallel planes S 0 x a2 . Characteristic quadratic form of quadric. The characteristic quadratic form F x y z aux2 a22y2 a33z2 2a12xy 2a13xz a23yz corresponding to equation and its characteristic equation an A a12 a13 a12 a13 a22 A a23 a23 a33 A 0 or A3 SA2 TA Ö 0 permit studying the main properties of quadrics. . Quadric Surfaces Quadrics 151 The roots A1 A2 and A3 of the characteristic equation are the eigenvalues of the real symmetric matrix aij and hence are always real. The invariants S T and 6 can be expressed in terms of the roots A1 A2 and A3 as follows S Ai A2 A3 T A1A2 A1A3 A2A3 6 A1A2A3. . Diameters and diameter plane. The locus of midpoints of parallel chords of a quadric is the diameter plane conjugate to these chords or the direction of these chords . The diameter plane conjugate to the chords with direction cosines cos a cos 3 and cos 7 is determined by the .
