Respuesta :
The standard form of an ellipse is
[tex] \dfrac{(x-x_0)^2}{a^2}+\dfrac{(y-y_0)^2}{b^2}=1 [/tex]
Let's try to complete some squares: if we focus on the part involing x, we have
[tex] 9x^2+54x = 9x^2+54x+81-81 = (3x+9)^2-81 = 9(x+3)^2-81 [/tex]
Similarly, for the part involving y, we have
[tex] 4y^2+8y = 4y^2+8y+4-4 = 4(y+1)^2-4 [/tex]
So, the equation becomes
[tex] 9(x+3)^2-81 + 4(y+1)^2-4 - 59=0 \iff 9(x+3)^2+ 4(y+1)^2-81-4-59=0 \iff 9(x+3)^2+ 4(y+1)^2 = 144 [/tex]
Divide both sides by 144 to get
[tex] \dfrac{(x+3)^2}{16}+ \dfrac{(y+1)^2}{36} = 1[/tex]
which is the standard form
