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The discriminant is a valuable tool for identifying the type of conic section an equation represents without graphing it. For a conic section equation in the form \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]the discriminant \( \Delta \) is calculated using the formula \[ \Delta = B^2 - 4AC \].

• If \( \Delta = 0 \), the conic is a parabola.
• If \( \Delta > 0 \), it is a hyperbola.
• If \( \Delta < 0 \), the conic is an ellipse.

For the given equation, \( \sqrt{3}x^2 + 3xy = 3 \), with \( A = \sqrt{3} \), \( B = 3 \), and \( C = 0 \), we find \( \Delta = 9 \). Since \( \Delta > 0 \), the graph is a hyperbola. Evaluating the discriminant in this way provides a quick method to classify and understand the nature of the conic section.

A hyperbola is a type of conic section that forms when a plane intersects both nappes of a double cone. Hyperbolas are characterized by two distinct, mirror-image curves called branches.

• Unlike ellipses or parabolas, hyperbolas have a transverse and a conjugate axis.
• The transverse axis passes through the vertices of the hyperbola.
• Asymptotes: Two lines that the curve approaches but never touches. They determine the direction of the branches.

In the Cartesian plane, hyperbolas appear as curved shapes with open ends extending towards infinity. These shapes are symmetric about their center. The primary features of a hyperbola include **vertices** (the closest points on each branch to the center) and **foci**, which lie inside each branch and determine its shape. The equation for a hyperbola can take on various forms, particularly when transforming or rotating the axes.

Sometimes equations of conics include an \(xy\)-term, which can complicate identification and graphing. To simplify, we can use a rotation of axes.

• The goal of this rotation is often to eliminate the \(xy\)-term from the equation.
• For rotation, utilize the formula \( \cot(2\theta) = \frac{A-C}{B} \) to find the rotation angle \( \theta \).

Using our example, where \( A = \sqrt{3} \), \( B = 3 \), and \( C = 0 \), we calculate \( \cot(2\theta) = \frac{\sqrt{3}}{3} \). Solving gives \( \theta = 15^\circ \). With this angle, we rotate using the formulas:- \( x = x'\cos\theta - y'\sin\theta \)- \( y = x'\sin\theta + y'\cos\theta \)This rotation results in a new equation without the \(xy\)-term, making it easier to understand and graph the conic shape. This refinement simplifies analysis and sketching, allowing for a clearer comprehension of the conic's structure. By using rotation, complex relationships transform into more recognizable forms.