Analyse the following equation using rotational axis

Question

4x^2 + 9y^2 - 8x + 36y + 4 = 0

Answer 1

I do not know what using rotational axis means but this is an ellipse

4(x^2-2x) +9(y^2+6y) = -4
complete squares

4(x^2-2x+1)-4+9(y^2+6y+9)-81=-4

4(x-1)^2 +9(x+3)^2 = 81

(x-1)^2/4.5^2 + (x+3)^2/3^2 = 1

ellipse