Problem 7 - Beginning and Intermediate Algebra

Question

Identify each equation as an ellipse or a hyperbola. $$\frac{x^{2}}{25}+y^{2}=1$$

Step-by-Step Solution

Answer

The given equation \(\frac{x^2}{25} + y^2 = 1\) is an ellipse, as both x and y terms are positive and it matches the general form of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Step 1: Identify the general form of the given equation

We are given the equation \(\frac{x^2}{25} + y^2 = 1\).

Step 2: Analyze signs of the terms

Look for the signs of the terms involving x and y. We have the equation \(\frac{x^2}{25} + y^2 = 1\). Both terms are positive as there is no negative sign before any of them.

Step 3: Compare the equation with the general forms of ellipse and hyperbola equations

The given equation has the form \(\frac{x^2}{a^2} + y^2 = 1\) which is the general form of an ellipse.

Step 4: Conclusion

The given equation \(\frac{x^2}{25} + y^2 = 1\) is an ellipse.

Key Concepts

Ellipse

An ellipse is a special type of conic section that looks like a stretched circle. Ellipses have several key properties that define their shape and dimensions. The defining feature of an ellipse is that the sum of the distances from any point on the ellipse to two fixed points, called foci, is constant. Ellipses can be oriented either horizontally or vertically.

• The standard form of an ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) if centered at the origin.
• Here, \(a\) and \(b\) are the lengths from the center to the ellipse along the x-axis and y-axis respectively.
• If \(a > b\), the ellipse is elongated along the x-axis. Conversely, if \(b > a\), it is elongated along the y-axis.

Understanding these elements can help you identify and work with ellipses when given an equation.

For example, the equation from the problem, \(\frac{x^2}{25} + y^2 = 1\), can be rewritten as \(\frac{x^2}{5^2} + \frac{y^2}{1^2} = 1\), indicating an ellipse stretched along the x-axis.

Hyperbola

A hyperbola is another type of conic section that appears as two separate curves or "arms". Unlike ellipses, hyperbolas have different properties and equations.

• In standard form, a hyperbola can be written as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\).
• The key distinction of hyperbolas is the presence of a subtraction between the terms, indicating the hyperbola's separate branches.
• Hyperbolas have two foci and a difference in distances from any point on the hyperbola to these foci, which is constant.

Hyperbolas often involve the concept of asymptotes, which are lines that the hyperbola approaches but never touches. It is important to recognize the minus sign in their equations, differentiating them from ellipses.

Equation Analysis

When analyzing conic section equations, identifying the signs and coefficients in the equation is crucial. The key aspect to look for is which conic form the equation resembles.

• If all the terms are positive, and it follows the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), then the conic is an ellipse.
• If there is a subtraction, \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\), it represents a hyperbola.

In our problem, analyzing the equation \(\frac{x^2}{25} + y^2 = 1\), we notice the terms are added, which is indicative of an ellipse. Identifying whether a conic section is an ellipse or a hyperbola based on their equations is a foundational skill in understanding conic sections.

General Form of Conic Sections

The general equation for conic sections provides a unified way of expressing ellipses, hyperbolas, circles, and parabolas. This form is versatile but identifying each conic section requires attention to specific characteristics within the equation.

• The general quadratic form is given by \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).
• Different values and combinations of \(A\), \(B\), and \(C\), as well as the absence or presence of certain terms, determine the type of conic.
• If \(B = 0\) and \(A, C > 0\) with the equation being positive, often it describes an ellipse.
• If \(B = 0\) and \(A, C\) have opposite signs, it represents a hyperbola.

By transforming or simplifying the general form equation, students can convert it into the standard form for each conic type, assisting in identifying the conic described. This understanding serves as a foundation for more complex problem-solving in conic sections.