ANM.YIK' OEOMETLY VIII 



that is, each has one and <>nly one term containing the 



if of a variable, and no term containing the product 



of the two variables. Conversely, it may be shown that 



an equation of either of these forms represents a parabola 



la parallel to one of the coordinate axes. 

 A numerical example will first be discussed, by the 

 method which has already been employed in connection 

 with the equation of the circle (Art. 79), and which is 

 applicable also in the case of the other conies. It is tin- 

 method of reducing the given equation to a standard form, 

 and is analogous to "completing the square" in the solu- 

 tion of quadratic equations. 



EXAMPLE. Given the equation 



25y a -30y-50* + 89 = 0, 



to show that it represents a parabola ; and to find its vertex, focus, and 

 directrix. 



Divide both members of the equation by 25, and complete the square 

 of the y-term> ; the Aquation may then be written 



that is, fo-!)* = 2(*-t), 



whence (y - $)* = * i'(* - !) 



Now this equation is in the second standard form (cf. equation [42]), 

 and therefore every point on its locus has the geometric proper iy uiv. n 

 in Art. 106; and the locus is a parabola. The vertex is at tin- point 

 (|,|); its axis is parallel to the z-axis, extending i" the po-itivi- <lir<-r- 

 li>:i ; and, since p = i, its focus is at the point ( ,. ), and the direm i\ 

 i- the linez= }. 



< "iisider now the general equation, and apply the same 

 method, taking for example the second form. \i/. : 



Ax* + 2GX + 2Fy + 0=0. 



I>i\idinLT loth numhcrs of the equation by A, completing the 

 square of the ar-ternis. and transposing, the equation becomes 



