We complete the square
[tex]4 x^2 + y^2 - 16x + 2y + 13 = 0[/tex]
[tex]4 x^2 - 16x + y^2 + 2y = -13[/tex]
[tex]4 (x^2 - 4x) + (y^2 + 2y) = -13[/tex]
[tex]4 (x^2 - 4x + 4) + (y^2 + 2y +1 ) = -13 +16 + 1 [/tex]
[tex]4 (x - 2)^2 + (y+1)^2 = 4 [/tex]
[tex]\dfrac{ (x - 2)^2 }{1^2} + \dfrac{ (y+1)^2}{2^2} = 1 [/tex]
That's standard form with axes 1 and 2
