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How can a quadratic equation in x and y be transformed into a standard conic-section equation?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

A general quadratic equation in two variables is expressed as $A x^{2} + B y^{2} + 2 C x y + 2 D x + 2 E y + F = 0$ . To transform this into a standard conic section equation, one must perform a suitable choice of coordinate axes.1

By applying coordinate transformations—specifically rotations to eliminate the $x y$ term and translations to shift the center or vertex—the equation can be reduced to one of three simple standard forms:1

Ellipse (including the circle): $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ 1
Hyperbola: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ 1
Parabola: $y^{2} = 2 p x$ 1

The specific shape of the curve is determined by the coefficients of the original equation, which can be analyzed using the discriminant and invariants to identify the conic or its degenerate cases. This algebraic approach to geometric curves is a fundamental component of Cartesian geometry, which allows for the systematic study of these shapes.21