Problem 309 - Calculus Volume 3

Question

Determine which of the conic sections is described. $$ x^{2}-x y+y^{2}-2=0 $$

Step-by-Step Solution

Answer

The conic section is an ellipse.

Step 1: Identify the General Form

The given equation is \(x^2 - xy + y^2 - 2 = 0\). We need to compare this equation with the general form of conic sections, which is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Here, \(A = 1\), \(B = -1\), \(C = 1\), \(D = 0\), \(E = 0\), and \(F = -2\).

Step 2: Compute the Discriminant

The discriminant \(\Delta\) for a conic section is given by \(\Delta = B^2 - 4AC\). Substituting the values, we have \(\Delta = (-1)^2 - 4(1)(1) = 1 - 4 = -3\).

Step 3: Determine the Type of Conic Section

Based on the discriminant, we classify conics as follows:- If \(\Delta = 0\), the curve is a parabola.- If \(\Delta > 0\), the curve is a hyperbola.- If \(\Delta < 0\), the curve is an ellipse.Since \(\Delta = -3 < 0\), the given equation describes an ellipse.

Key Concepts

Ellipse

An ellipse is a smooth, curved shape that resembles an elongated circle. It is defined by two main points known as foci. The sum of the distances from any point on the ellipse to these two foci remains constant.

When dealing with a conic section equation, identifying whether the equation represents an ellipse is crucial. Ellipses are a fundamental type of conic section, along with parabolas and hyperbolas. They are characterized by having a negative discriminant in their standard form equation.

Some noteworthy features of ellipses include:

• Major Axis: The longest diameter of the ellipse.
• Minor Axis: The shortest diameter of the ellipse.
• Eccentricity: A measure of how much the ellipse deviates from being circular; ellipses have an eccentricity between 0 and 1.

Understanding the properties of ellipses helps in both mathematical problems and real-world applications, such as orbits of planets and designing optical lenses.

Discriminant

The discriminant is a value calculated from the coefficients of a conic section's equation. It helps in determining the nature of the conic without solving the equation entirely.

For conic sections described in the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant \(\Delta\) is given by \(\Delta = B^2 - 4AC\).

Here's how the discriminant categorizes conics:

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

The discriminant is a powerful tool because it quickly tells us the type of conic section, just from the equation's coefficients. In this exercise, the calculated discriminant \(-3\) indicated an ellipse.

General Form

The general form of a conic section's equation is a structure that can describe any conic section: ellipse, parabola, or hyperbola. It looks like this:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\].

Each term in this equation plays a role:

• \(A\), \(B\), \(C\) influence the type and orientation of the conic section.
• \(D\), \(E\) move the conic section around the coordinate plane.
• \(F\) affects the size and position.

The general form is versatile, but it is often necessary to convert it into a more simplified, recognizable form to understand the specific type of conic section.

In our example, once the coefficients were identified, the discriminant could be computed to reveal that the equation was an ellipse. Understanding the general form lets us decode conic equations and identify the curves they represent.