ITS 



AN A I ) M/ /T/; T 





V 



given directrix and for us is in general not an simple a lard .--JM.-I- 



ti.m. It \\ill now be shown that if the coordinate axes be transform. .1 



so as to be parallel to the axis and 

 directrix of the curve, the equation \\ ill 

 1... n-diiri'd to a Mandard form. For ex- 

 ample, the equation of the parabola \\ it It 

 focus at (2, ~3), and having for lii < -i ri \ 

 tli- line x-2y-l=0, was found to be 

 4x + 4*y + y* - 18* -|- 26 y + 64 = 0. 

 The axis of the curve is a line through 

 ) and perpendicular to 



\ 



Fiu. 79. 



its equation is 2 x + y = 1, and it cuts 

 the ar-axis at the angle d = tan- J ( L'). 

 The point Z is the intersection of the directrix and axis, and may be 

 found from the two linear equations representing these lines ; the vertex 

 A is the point bisecting ZF. If, then, the axes are rotated through 

 the angle = tan- 1 (~2), the equation will be reduced to the <.. nd 

 standard form, [42] ; and if the origin be also removed to the vertex 

 A y the equation will be further reduced to the first standard form, [41]. 



The point Z is (f , ~|), A is (H ~S) 5 hence, p = A F = -^ , and trans- 

 forming the axes through the angle = tan-^-S), to the new origin 



(H '$) tne equation of the parabola reduces to y 2 = -x. 



v6 



The problem of reducing any equation representing a parabola to its 

 standard form is taken up more fully in Chap. XII. 



EXERCISES 



Find, and reduce to the first standard form, the equation of each of 

 the following parabolas ; also make a sketch of each figure : 



1. with focus at the point ( 1. 3), and having for directrix the line 



2. with focus at the point (-8, -J), and having for directrix the line 



o.r + 7 y _8 = 0; 



3. with focus at the point (a, 6), and having for directrix the line 



