Problem 27 - A Graphical Approach to College Algebra

Question

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$x^{2}=25+y^{2}$$

Step-by-Step Solution

Answer

The equation represents a hyperbola centered at the origin.

Step 1: Rearrange the Equation

Start with the given equation: \(x^2 = 25 + y^2\). To identify the type of conic section, rewrite the equation in a standard form by rearranging the terms. Subtract \(y^2\) from both sides to get: \(x^2 - y^2 = 25\).

Step 2: Identify the Standard Form

Compare the equation \(x^2 - y^2 = 25\) with the general form of a hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Here, \(a^2 = 25\) and \(b^2 = 25\), so the equation fits the form of a hyperbola.

Step 3: Determine the Type of Conic Section

Since the equation \(x^2 - y^2 = 25\) fits the standard form \(\frac{x^2}{25} - \frac{y^2}{25} = 1\), it represents a hyperbola. The hyperbola is centered at the origin \((0,0)\).

Step 4: Graph the Hyperbola

For the hyperbola \(\frac{x^2}{25} - \frac{y^2}{25} = 1\), the transverse axis is along the x-axis, and the vertices are at \((5,0)\) and \((-5,0)\). The conjugate axis is along the y-axis, with co-vertices at \((0,5)\) and \((0,-5)\). Sketch the hyperbola by plotting these vertices and drawing the asymptotes which are y = x and y = -x.

Key Concepts

Hyperbola

A hyperbola is one of the four main types of conic sections, formed by the intersection of a plane and a double napped cone. It is characterized by its distinct shape, which consists of two symmetrical open curves facing away from each other. The hyperbola is defined mathematically as the set of all points where the difference of the distances to two fixed points, called foci, is constant. This means each point on the hyperbola maintains an equal difference in distance between the two foci, creating its unique open shape. In real-life scenarios, hyperbolas can be observed in certain satellite dishes and radio waves, showcasing how their mathematical properties apply to practical applications.

Standard Form

The standard form of a hyperbola is vital for identifying its equation and graphing it correctly. It is written as either \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), depending on the orientation of the transverse axis. The important components in the standard form are:

• \(a^2\): represents the square of the distance from the center to each vertex.
• \(b^2\): is the square of the distance from the center to the co-vertices.
• \(c^2 = a^2 + b^2\): helps determine the distance to the foci, forming the equation of the hyperbola.

By rearranging the exercise equation \(x^2 - y^2 = 25\) into the form \(\frac{x^2}{25} - \frac{y^2}{25} = 1\), we ensure that it confirms the standard form of a hyperbola. This standard positioning aids in easy graphing by establishing a framework for the values of \(a\) and \(b\).

Transverse Axis

The transverse axis of a hyperbola is the axis along which the two symmetrical curves extend from the center. It is the longer axis in the hyperbola's orientation, providing the directional spread of the curve. For the equation \(\frac{x^2}{25} - \frac{y^2}{25} = 1\), the transverse axis is along the x-axis. This is because the \(x^2\) term comes first in the standard form.
The position of this axis determines how the hyperbola opens:

• If \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the transverse axis is horizontal.
• If \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the transverse axis is vertical.

By noting this axis, graphing becomes much more straightforward as it tells us the direction where the vertices and subsequent curves are placed.

Vertices

Vertices are essential points on a hyperbola, indicating where each of the two curves meets the transverse axis. For the standard form \(\frac{x^2}{25} - \frac{y^2}{25} = 1\), the vertices are at \((\pm5, 0)\). These points lie directly on the transverse axis and represent the closest distance between the two branches of the hyperbola.
To identify the vertices, it’s important to:

• Determine \(a\), calculated as the square root of \(a^2\), giving the distance from the center to each vertex.
• Remember that for the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the vertices are along the x-axis, while for \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) they would be along the y-axis.

Understanding where the vertices are will allow you to accurately sketch the hyperbola and its principal axis alignment. This knowledge is particularly useful for setting the parameters of the graph and ensuring each part of the hyperbola is correctly represented.