HELP PLS

OK....had to look up how to work with this and how to rotate a hyperbola....so I learned something new today!)

Re-write into form    a x^2 + b xy + c y^2 + dx + ey + f = 0

x^2 +4xy + y^2  -1 = 0

a) Hyperbola

b) cot 2phi =  (a-c)/b  =  (1-1)/4 = 0

       phi = pi/4

c)  x = x' cos pi/4   -  y' sin pi/4    =   (x'-y')/sqrt2

      y = x' cos pi/4  + y' sin pi/4    =   (x'+y'/sqrt2

Sub these into the original equation   x^2                               + 4xy                          + y^2                      = 1

                                                         1/2 (x'^2 - 2x'y' + y'2)    + 4 ( x'^2 -y'^2)/2   + 1/2(x'^2+2x'y' + y'^2) =1

                                                             3x^2 -y'^2 =1

See graph:

https://www.desmos.com/calculator/w9jpq2jzpj