Question

For the following equations, determine which of the conic sections is described. $$ x^{2}-x y+y^{2}-2=0 $$

Short answer

The equation describes an ellipse.

Step-by-step solution

Recall the Standard Forms of Conic Sections

Before analyzing the given equation, it's important to recall the general forms of conic sections. The standard forms are: - Circle: \(Ax^2 + Ay^2 + Dx + Ey + F = 0\) where \(A > 0\) is equal for both \(x^2\) and \(y^2\). - Ellipse: \(Ax^2 + By^2 + Dx + Ey + F = 0\) where \(A \, \text{and} \, B > 0\) and \(A eq B\). - Parabola: \(Ax^2 + Dx + Ey + F = 0\) or \(By^2 + Dx + Ey + F = 0\). - Hyperbola: \(Ax^2 - By^2 + Dx + Ey + F = 0\) or \(-Ax^2 + By^2 + Dx + Ey + F = 0\).

Identify the Coefficients in the Given Equation

The given equation is: \[ x^2 - xy + y^2 - 2 = 0 \]To match with the conic sections forms, identify the coefficients: - The coefficient of \(x^2\) is 1. - The coefficient of \(y^2\) is 1. - The coefficient of \(xy\) is -1. - The coefficients for \(x\) and \(y\) terms are both 0. - The constant \(F\) is -2.

Use the Discriminant to Determine the Conic Section

The discriminant of a conic section is given by \( B^2 - 4AC \), where \( A \) is the coefficient of \(x^2\), \( B\) is the coefficient of \(xy\), and \( C \) is the coefficient of \(y^2\). For the given equation: - \( A = 1 \)- \( B = -1 \)- \( C = 1 \)Calculate the discriminant: \[ B^2 - 4AC = (-1)^2 - 4(1)(1) = 1 - 4 = -3 \]

Interpret the Discriminant Result

The discriminant of a conic section differentiates them as follows: - If \( B^2 - 4AC > 0 \), the conic section is a hyperbola.- If \( B^2 - 4AC = 0 \), the conic section is a parabola.- If \( B^2 - 4AC < 0 \), the conic section is either a circle or an ellipse.Here, since \( B^2 - 4AC = -3 \), which is less than zero, the conic section is an ellipse due to \(A = C\).

Key concepts

Ellipse

An ellipse is one of the most recognizable conic sections that appears as a stretched circle. It can be described using its specific equation format. The general form for the equation of an ellipse is given by \( Ax^2 + By^2 + Dx + Ey + F = 0 \), where \( A \) and \( B \) are both positive, and crucially, \( A eq B \). This means that the coefficients in front of \( x^2 \) and \( y^2 \) differ in an ellipse, which results in the elongated shape.

In an ellipse, there are two focal points, and the sum of the distances from any point on the ellipse to each focus is constant. This geometric property differentiates ellipses from other conic sections.

If \( A = B \), the ellipse simplifies to a circle, since both axes have equal lengths. In the given problem you might notice that \( A \) and \( C \) are both 1, indicating that this represents a circle. However, in the context of discriminants and the given equation, it's classified under an ellipse due to the term \( xy \).

Discriminant

The discriminant is a valuable tool when dealing with conic sections because it helps to determine which specific shape, or section, you are dealing with. It is calculated using the formula \( B^2 - 4AC \), where:

• \( A \) is the coefficient of \( x^2 \)
• \( B \) is the coefficient of \( xy \)
• \( C \) is the coefficient of \( y^2 \)

This formula directly impacts the nature of the conic section:

• If \( B^2 - 4AC > 0 \), the conic is a hyperbola.
• If \( B^2 - 4AC = 0 \), it's a parabola.
• If \( B^2 - 4AC < 0 \), the conic could be an ellipse or a circle.

When solving the given equation, since the discriminant \( B^2 - 4AC = -3 \), which is less than zero, the equation represents an ellipse. This result shows us that despite the presence of the \( xy \) term, the structure maintains attributes representative of an elliptical shape.

Standard Forms of Conic Sections

Conic sections include circles, ellipses, parabolas, and hyperbolas. Each shape can be identified by its uniquely structured equation. Understanding their standard forms is key to analyzing and solving related mathematical problems.

Here are the standard forms:

• Circle: \( Ax^2 + Ay^2 + Dx + Ey + F = 0 \) where \( A > 0 \) and a circle is symmetrical, meaning \( x^2 \) and \( y^2 \) have the same coefficient.
• Ellipse: \( Ax^2 + By^2 + Dx + Ey + F = 0 \), where \( A > 0, B > 0 \), and \( A eq B \), providing the stretched round shape.
• Parabola: Typically written as either \( Ax^2 + Dx + Ey + F = 0 \) or \( By^2 + Dx + Ey + F = 0 \), which is open in one direction either horizontally or vertically.
• Hyperbola: This takes the form \( Ax^2 - By^2 + Dx + Ey + F = 0 \) or \(-Ax^2 + By^2 + Dx + Ey + F = 0 \), noted by its "saddle" shape, with opposing curves.

The standard forms provide a quick reference to classify any given equation into its respective conic section. With practice, recognizing these forms aids significantly in finding solutions and understanding the spatial properties of conics.