Explanation Given information : The equation of quadratic surface, x = y 2 4 − z 2 9 . Graph : The graph of the quadratic surface x = y 2 4 − z 2 9 can be sketched using the traces, For the plane x = 0 , the trace of the surfaces is, x = y 2 4 − z 2 9 0 = y 2 4 − z 2 9 z 2 9 = y 2 4 whichare intersecting lines in yz plane. For the plane x = k , the trace of the surfaces is, x = y 2 4 − z 2 9 k = y 2 4 − z 2 9 which is a hyperbola in the plane parallel to yz plane. For the plane y = 0 , the trace of the surfaces is, x = y 2 4 − z 2 9 x = 0 − z 2 9 − 9 x = z 2 which is a downward parabola in xz plane. For the plane z = 0 , the trace of the surfaces is, x = y 2 4 − z 2 9 x = y 2 4 − 0 4 x = y 2 which is a upward parabola in xy plane. Since, traces are intersecting lines in yz plane,a hyperbola in plane parallel to yz plane and parabolas in planes xz and xy , the graph of quadratic surface is provided below, Interpretation : The general equations of all quadratic surfaces along with the description of their traces is provided below, The standard equation of the sphere is x 2 + y 2 + z 2 = r 2 , where r is the radius of the sphere. The standard equation of ellipsoid is x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 , where a , b and c are constants. Ellipse shaped traces are found in coordinate planes. The standard equation of elliptic paraboloidis z = x 2 a 2 + y 2 b 2 , where a and b are constants. Parabola shaped traces are found in x z and y z coordinate planes

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 9.7, Problem 42PS

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To graph: The quadratic surface x=y24−z29 and identify the surface

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4. For each equation given below, write down the name of the surface it represents. The options are: plane, cylinder, sphere, ellipsoid, one-sheeted hyperboloid, two-sheeted hyperboloid, elliptic paraboloid, hyperbolic paraboloid, or elliptic cone. y² 5 7 (a) + z2 34 (b) )x² + 4y² = = 5 = 0 (c) (i) x² + y² − z² = 1 Answer: Answer: Answer:

1. Write an equation for the set of all points 5 units away from the x-axis. 2. The coordinates of point P are (2,3, 7). What is the exact distance between point P and the z-axis? 3. Write an equation whose graph is the set of all points whose distance from the yz plane is 9. 4. Write an equation for the sphere of radius 7 centered at (-6,4,-1). 5. Find a functionr(x, y, z) such that the level surface given by r=13 is the graph of f(x, y)=x° cos y-y² +20. 6. Let g(x,y,z)=6x² +3xz +5. The level surface given by g=8 is the graph of f(x, y). Find f(x,y). 7. Plane P contains the point (5,-4,-1) and is orthogonal to the vector i= 31 – 5j+2k. Find an equation whose graph is P. 8. Find the vector that is the projection of v = 2î +3j-4k onto w=-2i + j+3k (express your answer as a vector %3D resolved into components parallel to the coordinate axes, that is, in the form ai + bj+ck ). 9. Write an equation for the plane containing the points P(5,3,-4); Q(2,6,7); and R(8,4,-2). 10. Find a unit…

2. Use the method of completing the square to write the following equation in standard form and identify the surface: x²+4z2+8x-4y-16z+32=0.

Chapter 9 Solutions

Calculus

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To graph : The quadratic surface x = y 2 4 − z 2 9 and identify the surface

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