134 ANALYii' 9EOMMTET [Cn. vi 



18. Transform y* = 8z to new rectangular axes having tin- p.. mi 

 (18, 12) as origin, and making an angle cot - * :i wit h the old. 



19. Transform to rectangular coordinates, the pole and initial line 

 being coincident with the origin and x-axis, respectively: 



(a) p? = a'cos20, () p*cos20 = a, (y) p = *sinl>0. 



Transform to polar coordinates, the x-axis and initial lino being coin- 



,1 : 



20. (x* + y*)* = k*(x* - y*), the pole being at the point (0, 0) ; 



21. x* + y = 7 ox, pole being at the point (0, 0) ; 



22. x + y* = 16 x, the pole being at the point (8, 0). 



23. Transform the equation y a + 4 oy cot 30 4 ax = to an oblique 

 system of coordinates, wilh the same origin and x-axis, but the new 

 y-axis at an angle of 30 with the old x-axis. 



x a v a 



24. Transform the equation TH + Q = 1 t new axes, making the 



l'Mtive angles tan"" 1 ! and tan ~ J ( f), respectively, with the old x-axis, 

 the origin being unchanged. 



25. Transform the equation 



3x* + lOVJJxy - 7y 8 = (18 - 30\/3)x -f (42 + 30V3)y + (42 -f 90v/:'>) 



to the new origin (3, 3), with new axes making an angle of 30 with 

 the old. 



26. Transform the equation 3x a + 8xy - 3y = to the two straight 

 lines which it represents, as new axes. 



27. Transform - = 1 to the straight lines ~ - ^ = 0, as new 

 axes. 



28. Transform to polar coordinates the equation y*(2a - x) = x. 



29. Transform to rectangular coordinates the equation 



p = a (cos 20 + sin 20). 



30. Prove the formula for the distance in polar coordinates [1] by 

 transformation of the corresponding formula [2] in rectangular coordi* 



ta 



31. Transform the equation x cos a -f y sin a = p to polar coordinates. 





