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The discriminant is a very powerful tool in understanding the nature of conic sections. It helps determine whether a particular second-degree equation represents a parabola, ellipse, or hyperbola. In the case of the equation provided, which is expressed in the form \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\] we can calculate the discriminant using the formula \(D = B^2 - 4AC\). The values for \(A\), \(B\), and \(C\) are gathered from the expanded equation. Plugging in the values for this specific problem, we find\[D = 336^2 - 4(49)(576) = 112896 - 112896 = 0.\] Since the discriminant equals zero, the conic section is identified as a **parabola**.

• If \(D > 0\), the equation represents a hyperbola.
• If \(D < 0\), it represents an ellipse (or a circle, if \(A = C\)).

Rotating the axes is a technique used to eliminate the \(xy\)-term in a conic equation, making it easier to analyze and sketch the graph. The formula to find the angle of rotation, \(\theta\), is\[\cot 2\theta = \frac{A - C}{B}.\]This helps us determine the angle necessary to simplify our equation. For the problem, substituting the values from the expanded form,\[\cot 2\theta = \frac{49 - 576}{336} = \frac{-527}{336}.\] It takes some trigonometric maneuvering to find the value of \(\theta\). Once the angle is known, the rotation formulas applied are:

• \(x = x'\cos\theta - y'\sin\theta\)
• \(y = x'\sin\theta + y'\cos\theta\)

These transform the original \(x\) and \(y\) into new variables \(x'\) and \(y'\), which simplifies the equation by removing the \(xy\)-term, clarifying the analysis of the conic.

A conic equation takes the shape of a second-degree polynomial and defines the different types of conic sections such as circles, ellipses, hyperbolas, and parabolas. The general conic equation is given by:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.\]For simpler handling of these equations, especially when there's an \(xy\)-term present, it's sometimes beneficial to rotate the coordinate axes as mentioned earlier.

• Circles and ellipses have a negative discriminant (\(D < 0\)).
• Hyperbolas exhibit a positive discriminant (\(D > 0\)).
• Parabolas have a discriminant of zero (\(D = 0\)).

By expanding and identifying the coefficients \(A\), \(B\), and \(C\), along with utilizing the discriminant, you can discern the type of conic, understand its orientation, and graph it accurately. Studying these equations not only solves textbook problems but also broadens your mathematical perspective.