Question

What is standard form for an ellipse?

Answer 1

1. Aside from the ones in the usual form, there are a lot of ellipses. Take any quadratic equation of the form and solve it.
2. $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$
3. When the discriminant $B^2-4AC$ is negative and the constants A, B, C, D, E, and F are negative, the solutions form an ellipse. (There's also a condition on F to ensure that any solutions exist at all.) However, the ellipse does not have to be centred on the origin or have its axes aligned horizontally and vertically.
4. For instance, consider the equation $2x^2+3xy+7y^2+6x-5y+4=0$.
5. the ellipse's description You can move the plane in a stiff motion so that it is centred at the origin, which will change its equation. You can also rotate it such that its two axes align with the x and y axes. This alters the equation as well. Finally, you can divide by the constant by moving it to the other side of the equation, and the equation of the resulting ellipse will look like this $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
6. That's the standard form for the equation of an ellipse. The two axes of symmetry fall on the x - and y-axes and intersect them at $(\pm a,0)$ and $(0,\pm b)$.