Problem 6 - Intermediate Algebra

Question

Fill in the blanks. To write \(9 x^{2}-4 y^{2}=36\) in ______ form, we divide both sides by 36

Step-by-Step Solution

Answer

hyperbola

Step 1: Identify the Problem

We are given the equation \(9x^{2}-4y^{2}=36\) and we need to write it in a different form by dividing both sides by the appropriate value. This form is known as the standard form of a hyperbola.

Step 2: Divide Both Sides by 36

To write the given equation in a standard form, we need to divide every term in the equation \(9x^2 - 4y^2 = 36\) by the constant on the right-hand side, which is 36. This will help normalize the equation.\[\frac{9x^2}{36} - \frac{4y^2}{36} = \frac{36}{36}\]

Step 3: Simplify the Equation

Simplify each term in the equation after division:- \(\frac{9x^2}{36} = \frac{x^2}{4}\)- \(\frac{4y^2}{36} = \frac{y^2}{9}\)- \(\frac{36}{36} = 1\)Therefore, the equation simplifies to:\[\frac{x^2}{4} - \frac{y^2}{9} = 1\]

Step 4: Recognize the Standard Form

Recognize that the simplified form \(\frac{x^2}{4} - \frac{y^2}{9} = 1\) is the standard form of a hyperbola. The key characteristic of this form is that one term is positive and the other term is negative, indicating a hyperbola centered at the origin.

Key Concepts

Equation Normalization

The process of equation normalization is fundamental when transitioning an equation from its general form to its standard form. In the context of hyperbolas, this process starts by equalizing the equation to 1 by dividing every term by the constant on the right side of the equation. For instance, given the equation \(9x^2 - 4y^2 = 36\), we begin normalization by dividing each term by 36.

Normalization ensures that the equation takes the required form that is useful for identifying the type of conic section—like a hyperbola—and for simplifying further calculations. In practical terms, this action rearranges the equation to appear familiar and manageable, setting the stage for further simplification.

Remember that normalization isn't just about division; it sets the standard for comparing and analyzing equations in a consistent manner. This step is crucial because it allows different conic sections (e.g., ellipses, hyperbolas) to be easily distinguishable.

Simplify Algebraic Expressions

After normalization, the next step focuses on simplifying algebraic expressions in the equation. This step involves breaking down the equation to its simplest form by reducing fractions and combining like terms wherever possible.

For example, with the equation \(\frac{9x^2}{36} - \frac{4y^2}{36} = 1\), each term can be reduced:

• \(\frac{9x^2}{36}\) simplifies to \(\frac{x^2}{4}\)
• \(\frac{4y^2}{36}\) simplifies to \(\frac{y^2}{9}\)
• \(\frac{36}{36}\) simplifies to 1

The end result of these calculations is the simplified equation \(\frac{x^2}{4} - \frac{y^2}{9} = 1\), a more straightforward expression. Simplification is key because it aids in recognizing the characteristics of the hyperbola and in performing algebraic manipulations more efficiently. Always remember to double-check that each component properly adds up to maintain the equation's integrity.

Hyperbola Characteristics

Understanding hyperbola characteristics is essential when studying conic sections. The standard form of a hyperbola, such as \(\frac{x^2}{4} - \frac{y^2}{9} = 1\), reveals a lot about the hyperbola's structure and properties. This formula provides crucial insights into its orientation and size.

In this form, the positive term \(\frac{x^2}{4}\) and the negative term \(\frac{y^2}{9}\) suggest that the hyperbola opens horizontally. The denominators (4 and 9) are particularly insightful as they correspond to the squares of the distances from the center to the vertices (along x-direction for 4, and y-direction for 9).

By observing these characteristics, it becomes easier to graph the hyperbola and understand its geometric properties.

• The center is at the origin \((0,0)\)
• The transverse axis is along the x-axis, due to the positive \(x^2\) term
• The vertices are at \((\pm 2, 0)\)
• The co-vertices are not points on the hyperbola but help form the rectangle used to draw asymptotes

Understanding these characteristics aids in further studies or applications involving conic sections.