Problem 30 - Precalculus Mathematics for Calculus

Question

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$x^{2}-2 x y+3 y^{2}=8$$

Step-by-Step Solution

Answer

The conic is an ellipse, confirmed by the negative discriminant and graphing.

Step 1: Identify the coefficients

The given equation is \(x^{2} - 2xy + 3y^{2} = 8\). We can identify the coefficients for the quadratic terms: \(A = 1\), \(B = -2\), and \(C = 3\). These are coefficients of the terms \(x^2\), \(xy\), and \(y^2\) respectively.

Step 2: Calculate the discriminant

The discriminant \(D\) for a conic section is given by the formula \(D = B^2 - 4AC\). Substitute the values from Step 1: \(D = (-2)^2 - 4(1)(3) = 4 - 12 = -8\).

Step 3: Analyze the discriminant

The value of the discriminant \(D = -8\) is less than zero. For conics, this discriminant value indicates that the conic is an ellipse.

Step 4: Graph the conic

Utilize a graphing device or software to input the equation \(x^{2} - 2xy + 3y^{2} = 8\). Upon graphing, you will observe the shape of an ellipse, confirming the result obtained from the discriminant analysis.

Key Concepts

Discriminant of Conic Sections

The discriminant is a very powerful tool when it comes to identifying conic sections like circles, ellipses, parabolas, and hyperbolas. It involves using a part of the quadratic equation to determine the nature of the conic section. This is especially useful when visual representation isn’t readily available. In the general equation for conic sections, which is written as \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, \] the discriminant is calculated using the formula:\[ D = B^2 - 4AC. \]Depending on the value of \( D \), you can identify the type of conic:

• If \( D > 0 \), the conic is a hyperbola.
• If \( D = 0 \), the conic is a parabola.
• If \( D < 0 \), the conic is an ellipse or possibly a circle.

In this exercise, the discriminant \( D = -8 \) signifies an ellipse. Since \( D \) is negative, it corresponds to the shape of an ellipse, showing how this simple calculation can give powerful insights.

Ellipse Identification

Ellipses are geometrical shapes much like stretched circles, characterized by their elongated look. If a curve feels similarly continuous and loopy to a circle, but isn’t perfect, it's often an ellipse. An ellipse can be visualized when the discriminant \( D < 0 \). This is a direct indicator helping us separate ellipses from other conics like parabolas or hyperbolas. Another way to identify an ellipse is by considering the orientation of its axes. Generally, ellipses have two axes:

• The major axis: the longest diameter of the ellipse.
• The minor axis: the shortest diameter of the ellipse.

Ellipses are often seen in planetary orbits and architectural designs, bridging nature with mathematical precision.

Quadratic Equations in Two Variables

Quadratic equations form the foundation for understanding conic sections. In two variables, they turn into various curves upon graphing, such as circles, ellipses, hyperbolas, and parabolas. The general form of a quadratic equation in two variables is:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. \]Depending upon the coefficients of this equation, you'll determine the exact nature of the curve:

• Coefficient \( A \) is linked to the \( x^2 \) term.
• Coefficient \( B \) multiplies the \( xy \) term.
• Coefficient \( C \) corresponds to the \( y^2 \) term.

These coefficients play a crucial role in the equation's discriminant, a key player in identifying the exact type of conic section you are dealing with. Understanding these fundamentals makes tackling any conic-related problem much more systematic and intuitive.