CONIC SECTIONS, ANALYTICAL. 1 M 



tangents at C, D to the loci make with the axes of y, x re- 



Y.-ly, 



cot a + cot = cot {$ + tan 6 = -7 ^ . 



G98. A circle being traced on a plane, the locus of the vertex 

 of all cones on that base whose principal elliptic sections have an 

 eccentricity e, is the surface generated by the revolution about its 

 conjugate axis of an hyperbola of eccentricity e~ l , 



699. A straight line of given length slides between two 

 fixed straight lines, and from its extremities two straight lines 

 are drawn in given directions ; prove that the locus of their inter- 

 section is an ellipse. 



700. Two circles of radii r, r (r > r') touch each other, and a 

 conic is described having real double contact with both ; 



that, when the points of contact are not on different branches of a 



; l>ola, the eccentricity > ^ / (l 4- -?J , and the latus rectum 



is greater than, equal to, or less than r-r', according as the 

 conic is an ellipse, parabola, or hyperbola. If the contacts be on 



different branches of a hyperbola, e > - f , and the asymptotes 

 ays touch a fixed parabola, 



701. If ABC be a triangle circumscribing an ellipse, S t S 

 th. foot, -in.l if SA**SJi = SC, then will 



S'A.S'B.S'C . 



SA.SS.SC '' 



and each angle of the triangle ABC will lie between the values 



CO." if*. 



702. One angular point of a triangle, self conjugate to a 



prove that the circles on tin- opposite sides 



PS will have a common radical axis, which is normal at 



