- I ANALYTIC QEOMEiltY [Cii. IX. 



To turn the y-axis to tin* final | io-.it ion, making' an angli- 6 \\ith the 

 x-uxia, the equations for transformation an* (Art, ",":i. [25]), 



or, by equation (*J), 



' 



Substituting these values in equation (4), it becomes 



r 



or, dropping now the accents, 



vhirh is the required equation of the parabola. 



This equation may, however, be written more simply. Observing 



(Art. lo:5) that p( } + *\ is the focal distance of the new origin "', and 

 representing that distance by //, equation (0) becomes 



y'=4 7 /,. . . . [58] 



This equation is of the same form as equation (1), but is referred to 

 oblique axes. In general, therefore, the equation 



represents a parabola, and j is the distance of its focus from the origin. 



i.ition [58] states the following property for every point P of th? 

 parabola : 



a property entirely analogous to that of Art. 106. 



EXERCISES 



1. Find the diameter of y*= -~ s, \\hirh bisects the chords 

 to the line x - y + 2 = 0. 



2. A diameter of the parabola y s = 8ar passes through the j 

 ) ; what is the equation of its corresponding chord- ? 



3. Find the equation of the diameter of the parabola y* = 4 r I 

 which bisects the chords 2 y - 3 x = k. 



4. Find the equation of the tangent to the parabola (y-G)*=*(.r-K 

 which is perpendicular to the diameter y 4 = 0. 



