Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 57Techniques for Graphing Quadric SurfacesEllipsoids A sketch of an ellipsoidsx 2a + y22 b + z2= 1 (a > 0,b > 0,c > 0) (3.36)2 c2 can be obtained by first plotting the intersections with the coordinate axes, then sketchingthe elliptical traces in the coordinate planes.Notethat thecurve of intersection ofasurface with a plane iscalled the trace of thesurfacein the plane.Example 3.44 Sketch the graph of the ellipsoidSolution .........x 24 + y216 + z29 = 1Hyperboloids of One Sheet A sketch of a hyperboloid of one sheetx 2a + y22 b − z2= 1 (a > 0,b > 0,c > 0) (3.37)2 c2 can be obtained by first sketching the elliptical traces in the xy-planes, then the ellipticaltraces in the planes z = ±c, and then the hyperbolic curves that join the endpoints of theaxes of these ellipses.Example 3.45 Sketch the graph of the hyperboloid of one sheetSolution .........x 2 +y 2 − z24 = 1Hyperboloids of Two Sheet A sketch of the hyperboloid of two sheetsz 2c − x22 a − y2= 1 (a > 0,b > 0,c > 0) (3.38)2 b2 can be obtained by first plotting the intersections with the z-axis, then sketching the ellipticaltraces in the planes z = ±2c, and then sketching the hyperbolic traces that connectthe z-axis intersections and the endpoints of the axes of the ellipses.Example 3.46 Sketch the graph of the hyperboloid of two sheetsz 2 −x 2 − y24 = 1