Problem 55 - Algebra and Trigonometry

Question

In Exercises 49–56, identify each equation without completing the square. $$ 100 x^{2}-7 y^{2}+90 y-368=0 $$

Step-by-Step Solution

Answer

The given equation is of a hyperbola with its axis along the x-axis.

Step 1: Equation identification

The given equation can be rewritten in the general form of quadratic equation as: \(Ax^{2} + By^{2} + Cy + D = 0\) where \(A=100\), \(B=-7\), \(C=90\), and \(D=-368\). As \(AB < 0\), the graph of the equation represents a hyperbola.

Step 2: Comparing with standard hyperbola equation

The standard equation of hyperbola is \( \frac{{x^{2}}}{{a^{2}}} - \frac{{y^{2}}}{{b^{2}}} = 1 \) or \( \frac{{y^{2}}}{{b^{2}}} - \frac{{x^{2}}}{{a^{2}}} = 1 \). Here, as \(B < 0\), the equation is of the form \( \frac{{x^{2}}}{{a^{2}}} - \frac{{y^{2}}}{{b^{2}}} = 1 \), also known as a horizontal hyperbola.

Step 3: Final Conclusion

The given equation is a hyperbola with its axis along the x-axis (horizontal). It is important to note that while the coefficient \(C\) impacts the position, it does not change the overall shape of the graph. Therefore, can still be classified as a hyperbola.

Key Concepts

Equation Identification

When you're given a mathematical equation, one of the first steps is to identify what kind of equation it is. This allows you to understand the geometric shape it represents on a graph. In this exercise, the given equation is:

• \[ 100x^{2} - 7y^{2} + 90y - 368 = 0 \]

The goal is to identify this equation without going through the classical method of completing the square.
In the context of conic sections, identifying the type of conic section is possible through a unique feature in their equations. Look at the coefficients of \(x^2\) and \(y^2\). In this case, \(A=100\) and \(B=-7\).
The crucial factor here is that the product of coefficients of the squared terms, \(A\) and \(B\), is less than zero \((AB < 0)\). When this condition occurs, it indicates that the equation represents a hyperbola.

Quadratic Equation

A quadratic equation involves terms where the variables (like \(x\) or \(y\)) are raised to the power of two, making them key players in defining curves in graphing. In simpler terms, a quadratic equation will have terms like \(x^2\) or \(y^2\), and may also include linear terms like \(x\) or \(y\), along with constant terms.
The equation in the exercise, \(100x^{2} - 7y^{2} + 90y - 368 = 0\), is quadratic because it contains the squared terms \(x^2\) and \(y^2\).
The structure of a generic quadratic equation in two variables is \(Ax^2 + By^2 + Cx + Dy + E = 0\). Here, you can see that both \(A\) and \(B\) are non-zero, confirming it is indeed a quadratic equation in two variables.

• An important detail is that although this equation doesn't need the completion of the square for identification, completing the square is a handy tool to reshape quadratic equations into more standard forms when necessary.

This helps in easily identifying the nature of the graph represented by the equation.

Conic Sections

A conic section is a curve obtained by intersecting a cone with a plane. The intersection's nature depends on the angle at which the plane cuts through the cone.
Quadratic equations represent various conic sections depending on their coefficients and terms. The main categories are ellipses, circles, parabolas, and hyperbolas.
In this exercise, by identifying the identification criteria \(AB < 0\), we've discovered that the equation forms a **hyperbola**. Hyperbolas have characteristic shapes and unique properties that distinguish them from other conic sections.

• The equation for a hyperbola in its standard form might look like \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), which describes a horizontal hyperbola, evidenced by the negative coefficient of \(y^2\) in our case.
• Conversely, if the equation were \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\), it would describe a vertical hyperbola.

The path taken by hyperbolas is always open, representing two opposing curves extending indefinitely, contrasting the closed nature of ellipses and circles.