91Ó°ÊÓ

In the world of conic sections, the discriminant is a powerful tool that helps determine the type of conic represented by a given equation. For any conic section in the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant is defined as \[\Delta = B^2 - 4AC\].

The value of \(\Delta\) plays a crucial role in identifying the conic section:

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

Given the equation in our exercise, by substituting \(A = 21\), \(B = 10\sqrt{3}\), and \(C = 31\) into the discriminant formula, we calculate \(\Delta = (10\sqrt{3})^2 - 4(21)(31) = -2304\).

Since \(\Delta < 0\), the conic section is an ellipse. Understanding these basic properties of the discriminant allows students to efficiently categorize different conic sections.

The rotation of axes is a mathematical technique used to simplify conic equations by eliminating the \(xy\)-term. This is especially useful in transforming the equation of a conic section to align its axes with the coordinate axes.

The rotation involves changing the coordinate system using the formulas:
\[x = x' \cos \theta - y' \sin \theta\]
\[y = x' \sin \theta + y' \cos \theta\]
where \(\theta\) is the angle of rotation.

To find \(\theta\), use the equation \(\tan(2\theta) = \frac{B}{A-C}\). In our example, this calculates as \(\tan(2\theta) = -\sqrt{3}\), leading to \(2\theta = 120^\circ\) or \(\theta = 60^\circ\). However, to apply the rotation correctly, use \(\theta = 30^\circ\), which is half of \(60^\circ\).

Using \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\) and \(\sin(30^\circ) = \frac{1}{2}\), substitute these into the rotation formulas to derive a new axis-aligned equation. This process simplifies the original equation and directly leads to understanding the graph's orientation.

An ellipse represents a type of conic section characterized by the equation \(A'x'^2 + C'y'^2 = 144\) after rotation, as shown in the exercise. It's like a stretched circle and has unique properties regarding its geometric shape.

Ellipses have:

• Two axes: a major axis (the longest) and a minor axis (the shortest).
• A center point, which remains at the origin after rotation in our case.
• Focal points along the major axis, inside the boundary of the ellipse.

Once you have the simplified form of the equation, it becomes straightforward to identify the lengths of these axes. The major axis corresponds to the larger coefficient in the standard form.

In this example, the absence of linear terms indicates that the center is precisely at the axes' intersection. Drawing the ellipse involves plotting these axes and ensuring the graph reflects the calculated proportions. Visualizing an ellipse can be easier once it is reduced to its standard form through the rotation process.