'.] ///. ITT 





'-'(Ya 



{taring thin equation with tin- standard equation [ I ']. 



MHMI that iu IOCUH is a parabola, whoat paralh-1 



t" the y-axw, extending in the negative direction if A ami P 



like signs, an>l in th- j tinn if A ,in<l F have 



unlike signs. Ita vertex it) at the point f- , "^ j; 



y 



since j -- ua ia at tl 



') 



ii istlieline 



_'.!/ 



G* + F*-AC 



J.I/ 



NOTE. The transformation just given fails if A = or if F- < 

 in that case some of the terms in the last equation are infinite If. bow- 

 erer, A =0, the given equation becomes 2 Or + :/> + f = ; ami. thU 

 being of the first degree, represent* a straight line. If, on the other 

 band, F=0, the given equation reduces to A** + 2 Gx + C = 0, and repre- 

 sent two straight lines each parallel to the y-axis ; they are real and 

 distinct, real and coincident, or imaginary, depending upon the value of 

 G*- A C. These lines may be regarded as limiting forms of the parab- 

 ola (nee Chapter XII). 



EXERCISES 



Determine the vertex, focuft, latu* rectum, equation of the directrix 

 and of the axis for each of the following parabolas; also sketch each 



1 y- 5r+ \y - 10=0; 8, 5y - 1 = 8y* + 1 1 : 



f4-8 = 0; 4. f* + 2-l-Jr- 11=0. 



108. Reduction of the equation of a parabola to a standard form. In 

 it was shown that the equation of a parabola having any 



TAJC. AH. OEOM. 18 



