Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation represents a degenerate conic. Explain. $$16 x^{2}+25 y^{2}-32 x+50 y+16=0$$

Step-by-Step Solution

Answer

To determine whether our equation represents a degenerate conic, we need to rewrite the equation into standard form. If the equation shows that the conic could be degenerate, we then confirm or deny this by looking at the relationships between the coefficients in our standard form.

Step 1: Writing the expression in a complete square form

Rearrange the equation by grouping the x and y terms together, simplify and complete the square. This will transform the equation into a conic section format.

Step 2: Defining the standard form of the equation and identifying the conic section

After transforming the equation by completing the square process to its standard form, determine which conic section it is by referring to the standard forms of conic sections. If the value under the square root becomes zero, it signals a degenerate form.

Step 3: Verifying Degeneration

If the equation indicates it could be a degenerate conic, we will find a certain relationship between the coefficients in our standard form that show degeneration.

Key Concepts

Completing the Square

Understanding how to complete the square is essential for transforming quadratic equations and identifying the nature of conic sections. The process involves creating a perfect square trinomial from a quadratic expression, making it easier to solve or rewrite equations, especially when dealing with conic sections.

In the context of conic sections such as circles, ellipses, parabolas, and hyperbolas, completing the square is a technique used to convert the general quadratic equation into standard form. For a quadratic in one variable, say x, the steps are: group the x-terms, move the constant term to the other side, and then add and subtract the square of half the coefficient of the linear term inside the parenthesis to form a perfect square.For example, if we have a term like \( ax^2 + bx \), completing the square involves adding and subtracting \( (b/(2a))^2 \) inside the parentheses to form \( a(x + b/(2a))^2 \) and a remaining constant term outside the parentheses.

This method helps to easily identify and classify conic sections when analyzing their graphs algebraically.

Conic Sections in Algebra

Conic sections are curves that result from the intersection of a plane and a double-napped right circular cone. Algebraically, these curves can be described by second-degree polynomial equations or quadratic equations in two variables, x and y.

There are four basic types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each has a unique algebraic representation:

• Circle: \( x^2 + y^2 = r^2 \), where r is the radius of the circle.
• Ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where a and b are the lengths of the semi-major and semi-minor axes, respectively.
• Parabola: \( y^2 = 4ax \) or \( x^2 = 4ay \), where a is the distance from the vertex to the focus.
• Hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \), where a and b are related to the distances between the vertices and foci.

Recognizing these algebraic forms helps in the analysis of conic sections and in solving problems related to them, such as determining the nature of a conic from its equation.

Standard Form of Conic Sections

The standard form of a conic section is a rewritten version of its general quadratic equation in a way that reveals the type of conic and its properties, such as center, axes, vertices, and direction of opening for parabolas.

Transforming the general equation of a conic into its standard form often involves the method of completing the square, which had been discussed previously. The standard forms for conic sections are pivotal for distinguishing them, as well as for understanding their geometries and making graphing easier. Here are the standard forms:

• Circle: \( (x-h)^2 + (y-k)^2 = r^2 \)
• Ellipse: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
• Parabola: \( (y-k)^2 = 4a(x-h) \) or \( (x-h)^2 = 4a(y-k) \)
• Hyperbola: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) or \( \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \)

A degenerate conic occurs when the conic section becomes a point, line, or pair of intersecting lines, typically happening when the determinant of the conic's general equation equals zero. By putting the equation in standard form, it becomes evident when this degeneration occurs, helping identify such special cases in both theoretical study and practical problem-solving.