BOOK OF MATHEMATICAL PROBLEMS. 



1105. If a cone be described having a plane section of a 

 given sphere for base, and vertex at a point V on the sphere, the 

 subcontrary sections will be parallel to the tangent plane at V. 



1106. If a cone whose vertex is the origin and base a plane 

 section of the surface ax* + by 3 + cz* = 1 be a cone of revolution, 

 the plane must touch one of the cylinders 



(b-a)y a + (c-a)z*=l, (c-b)z* + (a-b)x*=l, 

 (a-c)x a +(b-c)y*=l. 



1107. A cone is described whose base is a given conic and 

 one of whose axes passes through a fixed point in the plane of 

 the conic : prove that the locus of the vertex is a circle. 



1108. If the locus of the feet of the perpendiculars let fall 

 from a fixed point on the tangent planes to the cone 



ax* + by* + cz* = 



i>e a plane curve, it will be a circle ; and in order that this may 

 be the case, the point must lie on one of the systems of straight 

 lines (of which only one is possible) 



a = 0) V +J !_ = o;Aa 



o a c a 



1109. Prove also that, if the point lie on one of these 

 straight lines, the plane of the circle will be perpendicular to the 

 other : and that a plane section of the cone perpendicular to one 

 of the straight lines will have one of its foci on that straight line 



and its eccentricity equal to /!* -~ H. 



1110. If a plane cut the cone ayz + bzx + cxy = in two 

 straight lines at right angles to each other, the normal to the 

 plane through the origin will also lie on the cone. 



1111. Prove that, when bb' = c'a', and cc' = a'b f , the equation 



ax*+ ... +2a'y+... + e 2a"x + ... +/= 



represents in general a paraboloid whose axis is parallel to the 

 straight line 



