CONIC SECTI"NS. ANALYTICAL. 



'\ Two parabolas have a common focus and axes in the 



:_rht line in ojijM^it.- .lirections; a circle is drawn through 



cus touching both parabolas ; prove that 



3r* = a*-(aay [ + a r *, 

 a, a' being the latera recta, and r the radius of the circle. 



G70. Through a fixed point is drawn any straight line, and 

 on it are taken two points, such that their distances from the 

 fixed point :ire in a constant ratio, and the line joining them 

 subtends a constant angle at another fixed point ; prove that 

 loci are circles. 



G71. If a triangle ABC circumscribe on ellipse, and its sides 

 subtend angles a, a'; ft /3'j y, y at the foci, 



sin (a - A) sin (a' - A) _ sin (/? - B) sin ((? - B) 

 sin a sin a' urn/? sin/? 



_ 



sin y sin/ 



the angles being taken so that a-f/J + y a' + /3' + y' = 2w. 



|g 

 C72. Four tangents to the parabola - = 1 + cos are drawn 



at the points 0,, f , 0^ 4 ; prove that the centres of the circles 

 nsMil.in^ th- l'..ur triangles formed by them lie on thu 



circle 



2r sinfl, sin g sin 0, sin 4 = a sin (0, + 0, + 0. + 4 - B). 



073. Two straight lines bisect each other at right angles; 

 t lint the locus of the points at which they subtend equal 

 is 



1 ing the lengths of the lines, th. : <>f intersection 



>le, and 2a the direction of the prime radius. 



07 L Two circles intersect, a straight line is drawn through 

 one of their common points, and tangents are drawn to the circles 



