CONIC SECTIONS, ANALYTICAL. 



Let the equations of the parabolas be y* = 4ax, y* = 4a* (x + a), 

 and let the three tangents be at w,, m s , m a . The point of inter- 

 section of (1) and (2) is - - , a ( + ) , and if this lie on the 



m i m v"i nt a ' 



second parabola 



or \ + ] = 

 wi jy 



and similarly for wi,, m a . Hence m v , w a are the two roots of the 



quadratic ill z, 



, /I 1 \ 8 4aa . , 

 a' I- + \ = + 4ao, 



JL_ ^ l_ 1 /4a' 2a*\ 1 1 4'a 



or 



> 9 m a ' \ ^i 



But if (A", Y) be the point of intersection of the tangents 



;2), (3), 



a /I 1\ 



+ I j 



... A'* =-' 

 a i 



and the equation of the locus is 



a parabola which coincides with tin- second if a = 4a'. 



549. Two parabolas have a common vertex A and a common 

 axis, an ordinate KPQ meeta them, and a tangent at P meets 



"liter parabola in It, J: . .I/.', AR' meet the ordinate in 

 Z, J/; prove that NP, NQ are re*j harmonic and geo- 



ic means between X /., .V.'/. 



550. A triangle is inscribed in a parabola and another 

 similar and simil uly situ.i- .inscribes it; prove that 



ides of i mgle are respectively four Umes the 



corresponding sides of the latter. 



