Finding the Equation of the Ellipse By Completing the Square

Question

I have the following equation of an ellipse: $$5x^2+9y^2+40x=100$$

I need to put it in the form: $$\frac{(x-h)^2}{m^2} + \frac{(y-k)^2}{n^2} =1 $$

I was trying to complete the square with the coefficients that have an $x$ variable.

$$5x^2 + 40x + 9y^2 = 100$$ $$5(x^2 + 8x) + 9y^2 = 100$$ $$5(x^2 + 8x + 16) + 9y^2 = 100 + 16$$ $$5(x+4)^2 + 9y^2 = 116$$

I then divided both sides of the equation by $116$.

$$\frac{5(x + 4)^2}{116} + \frac{9y^2}{116} = 1$$

However, when I graph the equation I do not get the same ellipse that was represented with the original equation of $5x^2 + 9y^2 + 40x = 100$. From the graph, it seems like they are two similar ellipses. Where am I making the mistake? Any help will be appreciated.

Answer 1

You did a good job just you made a mistake here $$5(x^2 + 8x + 16) + 9y^2 = 100 + 16$$ Should be $$5(x^2 + 8x + 16) + 9y^2 = 100 + 80$$ $$5(x^2 +4)^2 + 9y^2 = 180$$ $$\frac {(x^2 +4)^2}{36} + \frac {y^2}{20} = 1$$ $$.....$$

Answer 2

We have

$$5x^2+9y^2+40x=100\iff 5(x+4)^2+9y^2=180\iff (x+4)^2+\frac{y^2}{\frac59}=36\\\iff\frac{(x+4)^2}{36}+\frac{y^2}{20}=1$$