Graphing a Rotated Conic (a) Use the discriminant to...

Graphing a Rotated Conic (a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the xy-term. (c) Sketch the graph.
$$
\sqrt{3} x^{2}+3 x y=3
$$

Key Concepts

Graphing Rotated Conics

Graphing rotated conics involves two main steps: first, using a rotation of axes to remove the mixed xy-term, and second, rewriting the equation in one of the standard forms to identify key features like vertices, foci, and asymptotes. This process allows one to accurately sketch the conic and understand its orientation and position in the coordinate plane.

Rotation of Axes

Rotation of axes is a mathematical technique used to eliminate the xy-term in the equation of a conic. This is done by rotating the coordinate system by a specific angle, which simplifies the equation into one of the standard forms. The transformation involves using trigonometric relationships to redefine the coordinates, thereby making analysis and graphing more straightforward.

Discriminant Method

The discriminant in the context of conic sections is defined as B² - 4AC, where A, B, and C are the coefficients of the quadratic terms in the general second-degree equation. This value is used to classify the conic: if the discriminant is zero, the conic is a parabola; if it is negative, the conic is an ellipse (or a circle, if A equals C and B is zero); and if it is positive, the conic is a hyperbola.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The main types include parabolas, ellipses, and hyperbolas. Each type has its unique geometric properties and standard equations, which allow one to study their features such as vertices, axes of symmetry, foci, and directrices.

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