380 //;> [<-... n 



come relatively nearer. In fad, tin- hyj)rrlnliil may IM> 

 said to be tangent to the oone at infinity, and bears to 

 the cone a relation entirely analogous to that betv 

 the hypi'rbola and its asymptotes. In the same \\a\, 

 the cone s+- 0=0 is asymptotic to the hyperboloid 



EXAMPLES ON CHAPTER IV 



1. Derive the equation [35] directly fn.m UM> <l>finition of Ar 



2. Derive the equation [3<] directly from the definition of Ai : 



3. Derive the equations [:J7], [38] directly from the definitions of 

 Art. 1M1. 



4. Derive the equation [39] directly from the definition of Ar 



5. Show analytically that the intersection of two spheres is a circle. 



6. Find the equation of the tangent plane to the sphere (x-a)* 

 + (y - &) 2 + (z - c) a = r\ at any point of the sphere. 



7. Show that the equation A T^C -f 7?//,y + Cz^ + K = represents 

 a plane tangent to th<> quadric, Ax* 4- By* 4- C'z' 2 4- K = 0, at tin- point 

 C*j !/v i) on the quadric. 



8. Find the equation of the cone with origin as vertex and the ellipse 

 - + ^ = 1 in the plane z = - 2, as directrix. 



9. Find the equation of a sphere having the line from Pj= (r,, y r TJ) 

 to /= T c as a diameter. 



10. Show that a sphere is determined by four points in s; 



Write the equation of the quadric whose directing curves have the 

 equations : 



- = ,. and - 



14. 2 *=iez, and y* = 



15. x-4 = and c a 



