Problem 56 - A Graphical Approach to College Algebra

Question

Write an equation for each conic. Each parabola has vertex at the origin, and each ellipse or hyperbola is centered at the origin. $$\text { Focus }(2,0) ; e=\frac{6}{5}$$

Step-by-Step Solution

Answer

The equation of the hyperbola is \(\frac{9x^2}{25} - \frac{9y^2}{11} = 1\).

Step 1: Determine the Type of Conic

Given the eccentricity \(e > 1\), this conic is a hyperbola.

Step 2: Identify the Orientation

The focus is at \((2,0)\), which means the hyperbola opens horizontally along the x-axis.

Step 3: Use the Relationship Between Eccentricity and Foci

For a hyperbola centered at the origin with a focus at \((c, 0)\), the relationship is \(c = ae\). Given \(c = 2\) and \(e = \frac{6}{5}\), we calculate \(a\).

Step 4: Calculate the Value of \(a\)

Using the formula \(c = ae\), we have \(a = \frac{c}{e} = \frac{2}{\frac{6}{5}} = \frac{10}{6} = \frac{5}{3}\).

Step 5: Find the Value of \(b\)

For hyperbolas, \(c^2 = a^2 + b^2\). With \(c = 2\) and \(a = \frac{5}{3}\), compute \(b^2\).

Step 6: Calculate \(b^2\)

\(c^2 = 4\) and \(a^2 = \left(\frac{5}{3}\right)^2 = \frac{25}{9}\). Therefore, \(b^2 = c^2 - a^2 = 4 - \frac{25}{9} = \frac{36}{9} - \frac{25}{9} = \frac{11}{9}\).

Step 7: Write the Equation of the Hyperbola

The standard equation for a hyperbola opening horizontally is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Substitute \(a^2 = \frac{25}{9}\) and \(b^2 = \frac{11}{9}\) into the equation: \(\frac{x^2}{\frac{25}{9}} - \frac{y^2}{\frac{11}{9}} = 1\). Simplify: \(\frac{9x^2}{25} - \frac{9y^2}{11} = 1\).

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. Depending on the angle between the plane and the cone, different types of conics are formed. There are four main types:

• Circle
• Ellipse
• Parabola
• Hyperbola

In this specific problem, we are dealing with a hyperbola. A hyperbola is characterized by having two separate branches, which open in opposite directions. It is essential to understand the nature of conic sections because it allows us to write and manipulate their equations with ease. Knowing which type of conic section you are working with is foundational when solving problems or deriving their standard equations.

Eccentricity

Eccentricity (\(e\)) is a crucial concept when discussing conic sections. It determines the shape of the conic. The value of the eccentricity tells us which type of conic we have:

• If \(e = 0\), it's a circle.
• If \(0 < e < 1\), it's an ellipse.
• If \(e = 1\), it's a parabola.
• If \(e > 1\), it's a hyperbola.

In our exercise, the eccentricity is given as \(e = \frac{6}{5}\), which is greater than 1. Hence, the conic is a hyperbola. The larger the eccentricity, the more "stretched" the hyperbola appears. Eccentricity is not just a number; it is a measure of how much a conic section deviates from being circular.

Vertex

The vertex of a conic section is a point where the curve is most curved. For hyperbolas, there are two vertices, one for each branch. In our scenario, because the hyperbola is centered at the origin and opens along the x-axis, the vertices are symmetrically located on either side of the origin at \((\pm a,0)\).
For our hyperbola with \(a = \frac{5}{3}\), the vertices are at \((\frac{5}{3}, 0)\) and \((-\frac{5}{3}, 0)\). These points are crucial as they help to define the size and position of the hyperbola along with its axis of symmetry. Having an understanding of the vertex allows us to construct and visualize the hyperbola accurately.

Focus

A focus of a conic section is a special point used to define and construct the shape. In hyperbolas, there are two foci, one for each branch. The hyperbola itself is defined by the distances between any point on the hyperbola, the foci, and a constant difference.
For a hyperbola centered at the origin, the foci are located at \((\pm c, 0)\) if the hyperbola opens along the x-axis.
In our exercise, the focus is given at \((2, 0)\). Knowing the foci helps in understanding the spread and orientation of the branches of the hyperbola. The distance between the center and a focus, \(c\), is calculated using the relationship \(c^2 = a^2 + b^2\). This focus determines how "open" the hyperbola is, as a wider distance results in branches that open more.