Problem 17 - College Algebra

Question

For the following exercises, determine which conic section is represented based on the given equation. $$8 x^{2}+4 \sqrt{2} x y+4 y^{2}-10 x+1=0$$

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Identify the General Form

The given equation is \(8x^2 + 4\sqrt{2}xy + 4y^2 - 10x + 1 = 0\). This resembles the general form of a conic section equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), where \(A = 8\), \(B = 4\sqrt{2}\), \(C = 4\), \(D = -10\), \(E = 0\), and \(F = 1\).

Step 2: Calculate the Discriminant

The discriminant \(\Delta\) of the conic section is given by the formula \(\Delta = B^2 - 4AC\). Substitute the values: \(B = 4\sqrt{2}\), \(A = 8\), and \(C = 4\). We calculate \(\Delta = (4\sqrt{2})^2 - 4 \times 8 \times 4 = 32 - 128 = -96\).

Step 3: Analyze the Discriminant

The sign of the discriminant determines the type of the conic section: if \(\Delta < 0\), it represents an ellipse; if \(\Delta = 0\), it represents a parabola; if \(\Delta > 0\), it represents a hyperbola. Since \(\Delta = -96 < 0\), the equation represents an ellipse.

Key Concepts

Ellipse

In the world of geometry, an ellipse is a fascinating shape that resembles a stretched circle. You can think of it as an oval that is drawn around two points known as foci. Each point inside an ellipse has a unique property: the sum of the distances from any point on the ellipse to these two foci is constant. This geometric property makes ellipses very special and different from other conic sections. Ellipses occur naturally in various scientific contexts. For instance:

• The orbits of planets around the sun are elliptical.
• Sound waves can form ellipses as they bounce off surfaces.

Another interesting fact about ellipses is their axes. They have a major axis, which is the longest diameter, and a minor axis, which is the shortest diameter. These axes intersect at the center of the ellipse. The more elongated an ellipse is, the closer its shape approaches that of a line, whereas the more rounded it appears, the more it resembles a circle.

General Form of Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. These intersections result in four types of curves: circles, ellipses, parabolas, and hyperbolas.The general form of a conic section is given by the equation:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]where:

• \(A\), \(B\), and \(C\) are the coefficients of the squared terms.
• \(D\), \(E\), and \(F\) are the coefficients of the linear terms and a constant term.

This general form encompasses all types of conic sections. By analyzing the coefficients in the equation, it's possible to determine which kind of conic section it represents. This form allows us to easily calculate the discriminant, which will help us further identify the type of conic section.

Discriminant Analysis

Discriminant analysis is a crucial step in classifying conic sections from their general equation. The discriminant, denoted as \(\Delta\), is calculated using the formula:\[ \Delta = B^2 - 4AC \]The value of \(\Delta\) indicates the type of conic section:

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

In the exercise solution provided, the discriminant was calculated as \(-96\). With \(\Delta\) being less than zero, it indicates that the conic section is indeed an ellipse. Discriminant analysis provides clear, analytical insights into understanding which conic section is present from the general equation, showcasing the beauty and precision of mathematical analysis.