Problem 1 - Intermediate Algebra

Question

Identify the graph of each equation as an ellipse or a hyperbola. Do not graph. \(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\)

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Analyze the equation structure

The given equation is \(\frac{x^2}{16} + \frac{y^2}{4} = 1\). We notice that both \(x^2\) and \(y^2\) are positive and added together in the equation. This is a key characteristic of an ellipse.

Step 2: Confirm ellipse characteristics

In the standard form of an ellipse centered at the origin, the equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). In this equation, \(a^2 = 16\) and \(b^2 = 4\). Both denominators are positive, which is consistent with the equation of an ellipse.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These intersections create different shapes depending on the angle and position of the plane. The four primary types of conic sections are:

• Circle
• Ellipse
• Parabola
• Hyperbola

Each of these shapes has specific characteristics and equations. For instance, an ellipse forms when the intersecting plane cuts through both nappes of the cone at an angle less steep than the side of the cone but does not pass through its base.
An important aspect of conic sections is that they can all be expressed by diverging from the general quadratic equation:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]By adjusting parameters, different conic sections emerge. Recognizing these conic sections is crucial in understanding their unique properties and applications in fields like physics, astronomy, and engineering.

Equation of an Ellipse

The equation of an ellipse plays a central role in defining its structure. An ellipse is the set of all points such that the sum of the distances from two fixed points, called the foci, is constant.
In standard form, an ellipse centered at the origin is represented by:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]Where:

• \(a^2\) and \(b^2\) are the squares of the lengths of the semi-major and semi-minor axes, respectively.
• If \(a > b\), the ellipse is stretched along the x-axis, making it horizontal.
• If \(b > a\), it is elongated along the y-axis, making it vertical.

This equation contrasts with the equation of a hyperbola, where terms are subtracted. Understanding these differences is vital to correctly identifying and working with these geometric figures.

Standard Form of Ellipse

The standard form of an ellipse provides an easy way to identify whether a given quadratic equation represents an ellipse. The standard form is given as:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]This form is clear because it highlights the important features of the ellipse, namely its radii, or axes.

Here's what to look for in the standard form of an ellipse:

• The terms \(x^2\) and \(y^2\) must be positive.
• The equation equals 1, ensuring a normalized representation.
• The denominators \(a^2\) and \(b^2\) determine the lengths of the semi-major and semi-minor axes.

An ellipse's dimensions and orientation can be quickly understood from its equation by identifying \(a\) and \(b\). If both are equal, it represents a circle, which is a special kind of ellipse. This simplicity in form makes it easier to analyze and graph ellipses for various applications.