Problem 28 - Precalculus: Mathematical 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 graphing.

Step 1: Identify the coefficients

Write down the general form of the conic section equation, which is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). For the given equation \(x^2 - 2xy + 3y^2 = 8\), identify the coefficients: \(A = 1\), \(B = -2\), \(C = 3\), \(D = 0\), \(E = 0\), and \(F = -8\).

Step 2: Calculate the discriminant

The discriminant formula for a conic is \(B^2 - 4AC\). Substitute the identified coefficients: \((-2)^2 - 4(1)(3) = 4 - 12 = -8\).

Step 3: Analyze the discriminant

Determine the type of conic section using the discriminant. If \(B^2 - 4AC < 0\), it indicates an ellipse. Based on the discriminant \(-8\), which is less than zero, the given conic section is an ellipse.

Step 4: Confirm by graphing

Use a graphing device to graph the equation \(x^2 - 2xy + 3y^2 = 8\). Observe that the graph resembles an ellipse, thus confirming our discrimination analysis.

Key Concepts

Discriminant in Conic Sections

In conic sections, the discriminant is a crucial tool for identifying the type of conic section represented by a quadratic equation. For an equation in the general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant is given by the formula \(B^2 - 4AC\). This formula helps to determine the nature of the conic

• If \(B^2 - 4AC > 0\), the conic is a hyperbola.
• If \(B^2 - 4AC = 0\), the conic is a parabola.
• If \(B^2 - 4AC < 0\), the conic is an ellipse, or a circle if \(A = C\) and \(B = 0\).

By using the discriminant, we confirmed that the equation \(x^2 - 2xy + 3y^2 = 8\) describes an ellipse, as the discriminant \(-8\) is less than zero.
Understanding what the discriminant indicates allows us to quickly and effectively identify the conic, aiding us when graphing it later on.

Understanding Ellipses

An ellipse is a type of conic section that looks like an elongated circle. The defining feature of an ellipse is that it has two main axes: the major axis and the minor axis.

• The major axis is the longer diameter across the ellipse.
• The minor axis is the shorter diameter across the ellipse.
• The points where the axes meet are the center of the ellipse.

To understand ellipses better, imagine a circle stretched in one direction, which forms an oval shape. This shape can be described mathematically by its equation. An ellipse centered at the origin can be written as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the semi-major and semi-minor axes respectively. Note that if \(a\) equals \(b\), the ellipse is actually a circle. Recognizing an ellipse from its equation helps with visualizing and graphing it correctly based on its properties.

Graphing Conic Sections

Graphing conic sections, like ellipses, requires an understanding of their equations and properties. With today’s technology, graphing devices can simplify this task considerably. When you input the equation \(x^2 - 2xy + 3y^2 = 8\) into a graphing calculator or software, it transforms the equation into a visual format.
Here are tips to graph conics effectively:

• Understand the general shape the conic will take based on the discriminant.
• Use the equation to find key features like vertices, axes, and center.
• Ensure the graphing tool settings accommodate variations in the conic’s scale.
• Analyze the plotted graph to identify if the shape matches expectations, confirming calculations.

By graphing the conic sections, students gain a deeper insight into how these shapes translate from equations to geometric forms. This visualization helps in confirming the conic type determined algebraically, closing the loop between theoretical math and practical application.