352 ANALY1K BfQMfm [CH.IL81& 



The paraboloid of revolution : the surface generated by tint 

 parabola x* = 4 /?z, rotated about its axis, 



EXAMPLES ON CHAPTER II 

 What is the locus of each of the following equations? 



1. x*-6x + 9 = 0. 4. ax* + bxy + cy* = 0. 



2. 2 1 + 4 = 0. 5. I //; + 6y - 82 + 1 = 0. 



3. r'-2* + +2x-2 + l=0. 6. 2*-9 = 9. 



What are the curves of intersection of the surfaces represented by 

 the equations 



7. y + 1 = 0, 3z a + 3y a + 3c = 20? 



8. .r 2 y* = 0, 2 = 0? 



10. 9 (z 2 + y 2 ) - 2 a = 25 10 2, z = 5? 



Determine the traces upon the coordinate planes of the following 

 surfaces : 



12. z 2 + y + 42* = 25 ; 13. 3z 2 - 4y 2 - 2 2 = U. 



Find the equation of 



14. the paraboloid of revolution one of whose traces is y a = 5z -f 3. 



15. the cone of revolution one of whose traces is y = 5 * + 3 and 

 whose axis is the axis of y. Find its vertex. 



16. the oblate spheroid one of whose traces is -f = 1. 



2 3 



V 2 2 a 



17. the prolate spheroid one of whose traces is ^ + = 1. 



I V 



18. the surface of revolution whose axis is the axis of x and one of 

 whose traces is x 2 y 1=0. 



19. the hyperboloid of two nappes one of whose traces is 

 r-9* = l. 



20. the sphere described about the major axis of the ellipse 



as diameter. 



