?.] TRANSFORM A I C00JUM.N .1 / / - LSI 



10. Through what angle most the axes be turned in order that the 

 new equation of the line fl x + 4 j = 24 shall have no x-termT Show 

 analytical! v (cf. also examples 8 and 0). 



801 Ut be the required angle; then the equations of traus- 



formatiou are 



x = x* cos - / sin and y = x 7 sin + / co* 0; 



the new equation U 



(600*0 + 4sin0)x / - (6sin0 - 4co0)/ 94; 

 it is required that the coefficient of * be 0, 



6cos0 + 4sin0 = 0, U n tan0 = - f; 



the equation 



(6sin0-4oo 



which reduce* to ^/ + 24 



U. Through what angle must the axes be turned to remove the 

 : um the equation of the locus Ax + Jiy+C = Qt to remove 

 they-term? 



12. Show that to remove the xy4erra from the equation of the locus, 

 * - 5xy - 8 f* = 8 (cf. Ex. 5), the axes must be turned through the 

 angle = 67 3(X, i.s so that tan 20 = -1. What is the new equation? 



13. Through what angle must a pair of rectangular axes be turned 

 that the new x-axis may pass through the point ( -2, -5)? 



14. What point must be the new origin, the direction of axea being 

 unchanged, in order that the new equation of the line Ax + By +C = Q 

 hall have no constant term? 



15. To what point, as origin of a pair of parallel axes, must a trans- 



of axes be made in order that the new equation of the locus, 

 ay-y'-x + ysO, shall have no terms of first degree? Construct the 

 locus. 



16. Find the new origin, the direction of axea remaining unchanged, 

 so that the equation of the locus, x* + xy-3x-y + 2 = 0, shall have 

 no constant UTIM. ( . -mtruct the figure. 



17. Transform the equation 4x* + 2v / 5xy + 2*" = 1 to new rectan- 

 gular axes making an angle of 80 with the given axes, origin unchanged. 



