Deriving equation of ellipse from expanded form?

Question

The equation of an ellipse centered around the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ The expanded form is $Ax^2 + By^2 + Cx + Dy + E = 0$ How do I derive the second from the first? I have completed the square many times yet I still cannot algebraically manipulate it into the first equation above. I feel hopeless as I am studying mathematics and eventually want to get into higher topics and this simple problem has stumped me :( Thanks!

Answer 1

It seems to me that if you’re comfortable with completing the square, you shouldn’t have had any difficulty. Here’s how I work: $$ Ax^2+Cx=A\left(x^2+\frac CAx\right)=A\left(\left(x+\frac C{2A}\right)^2-\frac{C^2}{4A^2}\right)=A\left(x+\frac C{2A}\right)^2-\frac{C^2}{4A} $$ Do the same thing for the $y$-terms, throw all the constants over onto the righthand side of the equation, and get $$ A(x-a)^2+B(y-c)^2=\Psi\,, $$ and then you divide both sides by $\Psi$, take suitable square roots, and there you are.