write equation in terms of rotated x' y' system using theta, 
the angle of rotation. Write the equation involving x' y' 
in standard form.

; θ = 45°

If a locus is defined on the xy-coordinate system as , then 
it is denoted as on the rotated x'y'-coordinate system.

We calculate the substitutions to make:

x = x'cosθ-y'sinθ,  

y = x'sinθ+y'cosθ

x = x'-y' = (x'-y')

y = x'+y' = (x'+y')

x² = (x'-y')² = (x'²-2x'y'+y'²}=(x'²-2x'y'+y'²}

xy = (x'-y')(x'+y')=(x'-y')(x'+y')=(x'²-y'²)

y² = (x'+y')² = (x'²+2x'y'+y'²}=(x'²+2x'y'+y'²}



·(x'²-2x'y'+y'²}-·(x'²-y'²)+·(x'²+2x'y'+y'²}-72=0

·(x'²-2x'y'+y'²}-·(x'²-y'²)+·(x'²+2x'y'+y'²}-72=0

Clear of fractions by multiplying through by 2:

 13(x'²-2x'y'+y'²}-10(x'²-y'²)+13(x'²+2x'y'+y'²}-144=0

13x'²-26x'y'+13y'²-10x'²+10y'²+13x'²+26x'y'+13y'²-144=0

                                      16x'²+36y'²-144 = 0
Divide through by 4

                                       4x'²+9y'²-36=0
                                       
                                          4x'²+9y'²=36
 

Divide through by 36:

                                   
                                    

                                    





Edwin