Chapter 1 Topics in Analytic Geometry

•••MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 56P•0Theorem 3.16 The distance D between a point P 0 (x 0 ,y 0 ,z 0 ) and the plane ax+by+cz+d =0 isD = |ax 0 +by 0 +cz 0 +d|√a2 +b 2 +c 2 (3.34)Example 3.41 Find the distance D between the point (1,−4,−3) and the planeSolution .........Example 3.42 The planes2x−3y +6z = −1x+2y −2z = 3 and 2x+4y −4z = 7are parallel since their normals, 〈1,2,−2〉 and 〈2,4,−4〉, are parallel vectors. Find thedistance between these planes.Solution .........Example 3.43 It was shown in Example 3.29 of Section2.5 that the lineare skew. Find the distance between them.Solution .........3.7 Quadric SurfacesQuadric SurfacesL 1 : x = 1+4t, y = 5−4t, z = −1+5tL 2 : x = 2+8t, y = 4−3t, z = 5+tIn the discussion of Formula (1.11) in Section 1.2 we noted that a second-degree equationAx 2 +Bxy +Cy 2 +Dx+Ey +F = 0represents a conic section. The analog of this equation in an xyz-coordinate system isAx 2 +By 2 +Cz 2 +Dxy +Exz +Fyz +Gx+Hy +Iz +J = 0 (3.35)which is called a second-degree equation in x, y, and z. The graphs of such equationsare called quadric surfaces or sometimes quadrics.Six common types of quadric surfaces are ellipsoids, hyperboloids of one sheet,hyperboloids of two sheet, elliptic cones, elliptic paraboloids, and hyperbolicparaboloids.