i>.| /// 



'ont* can be drawn to the parabola. If, however, /" w 

 on the curve, then y* 4 jut 0, the two value* of m are 



;enU coincide. If /' .side the 



paraln'U, thru y f> 4 j>y < 0, and the two value* of m are 



o are no real tangent line*. Then 

 only \\ oiitM.l.- tiio parabola th.it two real and 



.;.-m line* may be drawn from it to the |)arabola. 



(4) Throwjh any point in the plant three normal* can be 

 drawn to the parabola. 



in. y-mx-2pm-pnt (1) 



is normal to the parabola y 1 = 4 y/ for all value* of m 



If P f a(x f ,y t ) lie any point ,,f th.- plan.-, then 



tli.- nrm,il (1) will pass through J'' if, and only if, m has a 



value that will s. I-.JM.II 



y a/m - 2pm - pm*. . . (2) 



.1 ml. n- in m, there are three values of 

 whirii :ii,- given conditions, and therefore, in gen- 



three normal* may be drawn to a parabola from a giv.-n 

 jMM-ial cases may, however, arise in whirh two of 

 root* of equ re equal, \\lim tht-n- would be 



two ilitlcn-nt niMin.tl IIIK-S; or all the roots may be 

 ual,* v and one real, in both of whit h 



there would U; onlv one normal line. Through e\ 

 I. ist on normal line can be drawn to the parabola. 



'tt9 at the ertrrmitie* of a focal -/i.r./ int^reect 

 the tlirfct, \ ,,t ri : /l,f <in./l,-* ( )elow). 



, if Ss^.y') is tin* jNiint of intT*M-tion of th.- 

 ts at th. . \tn tuitirs .f the focal chord, tlu-n the chord 

 o polar of , an.l itx ,-,|u. 



vU.: / v a(ip,0) ( areaUtlMroouof eqoailoo (X) eqiuL 



