ANALYTH GEOMETRY 



[CM. IX 



The abscissa of tin- point of contact of the loci of equa- 

 tions ( -) and [56] may be found from equation (3), by snl. 



stituting in it the value of k given in equation (4); it is -. 



tir 



Tin- Midinate may then be found from equation (1); it i- 



. The point of contact is then ( 'L ? ). 

 m \rw 2 m J 



136. The equation of the normal to the parabola y* = 4 / . 

 in terms of its slope. Since, by delinition, tin normal to a 

 curve is prrpmdirnlar to the tangent at the point of con- 

 tact, the equation of a normal to the parabola 



y* = \px (I, 



is, if m' be the slope of the tangent [Arts. 62, 18." ]. 



If m l>e the slope of the normal, then 



I 



m' 

 and equation (2) may be written 



y = mx 2pm pm*. . . . [~">7 

 This is the equation of a normal in terms of its .\vn 

 slope m. 



137- Subtangent and 



F ^ subnormal. Constructioi 



of tangent and normal. 

 Let P l = (a? r y^) 

 any given point on 



_X parabola whose eqi 



tion is 



Fio.lji. 



Draw the ordii 

 MP V the tangent TP l 



and the normal 



