Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes. \(\frac{x^{2}}{25}-\frac{y^{2}}{36}=1\)

Step-by-Step Solution

Answer

Vertices: (5, 0), (-5, 0); Foci: (√61, 0), (-√61, 0); Asymptotes: y = ±6/5x.

Step 1: Identify the standard form of the hyperbola

The given equation is \( \frac{x^{2}}{25} - \frac{y^{2}}{36} = 1 \). The standard form of a hyperbola with a horizontal transverse axis is \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \). It matches the given equation, where \( a^{2} = 25 \) and \( b^{2} = 36 \).

Step 2: Calculate the vertices of the hyperbola

Vertices of a hyperbola with the form \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) are \( (\pm a, 0) \). Here, \( a = \sqrt{25} = 5 \). Thus, the vertices are \( (5, 0) \) and \( (-5, 0) \).

Step 3: Calculate the foci of the hyperbola

The formula for the foci of a hyperbola in this form is \( (\pm c, 0) \), where \( c = \sqrt{a^{2} + b^{2}} = \sqrt{25 + 36} = \sqrt{61} \). Hence, the foci are \( (\sqrt{61}, 0) \) and \( (-\sqrt{61}, 0) \).

Step 4: Write the equations of the asymptotes

The equations of the asymptotes for a hyperbola of this type are \( y = \pm \frac{b}{a} x \). Substituting \( b = 6 \) and \( a = 5 \), the asymptotes are \( y = \pm \frac{6}{5} x \).

Key Concepts

Equation of Hyperbola

A hyperbola is a fascinating curve, and its equation helps us understand its structure. In the given equation \( \frac{x^{2}}{25} - \frac{y^{2}}{36} = 1 \), we can see it matches the standard form of a hyperbola with a horizontal transverse axis.

This form is expressed as \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), where \( a^{2} \) and \( b^{2} \) represent the denominators of the respective terms. Here, \( a^{2} = 25 \) and \( b^{2} = 36 \).

Understanding this standard form is crucial as it lays the foundation for finding other properties like vertices, foci, and asymptotes. Knowing \( a^{2} \) and \( b^{2} \) helps us calculate essential values that further describe the hyperbola.

Vertices and Foci

Vertices and foci are key features of a hyperbola that define its size and shape. For the standard form \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), vertices are located at points \( (\pm a, 0) \).

In the given equation, \( a = \sqrt{25} = 5 \), so the vertices are positioned at \( (5, 0) \) and \( (-5, 0) \).

The foci lie further away from the center along the axis than the vertices, and their calculation involves \( c = \sqrt{a^{2} + b^{2}} = \sqrt{25 + 36} = \sqrt{61} \). Thus, the foci are \( (\sqrt{61}, 0) \) and \( (-\sqrt{61}, 0) \). Understanding where these points are helps visualize the hyperbola's orientation and its extension in space.

Asymptotes of Hyperbola

Asymptotes are imaginary lines that the hyperbola approaches but never actually crosses. They provide a framework that the hyperbola follows, giving an insight into its widening nature as it extends to infinity.

The asymptotes for a hyperbola like \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) are represented by equations \( y = \pm \frac{b}{a} x \).

For our specific equation, substituting the calculated values gives us \( y = \pm \frac{6}{5}x \). These lines guide the general shape and open direction of the hyperbola, allowing us to anticipate its growth beyond the typical graphing window.

Standard Form of Hyperbola

The standard form of a hyperbola’s equation allows us to quickly determine its fundamental attributes such as axes orientation and dimensions.

This crucial form, \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), describes a hyperbola that opens horizontally, while \( \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1 \) describes one that opens vertically.

The denominators, \( a^{2} \) and \( b^{2} \), are essential in this form as they help reveal the direction and spread of the hyperbola. Using this format helps simplify the task of identifying properties like vertices and foci, ensuring an accurate understanding of this complex geometrical figure.