Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

Find an equation of the hyperbola with foci \((h \pm c, k)\) and vertices ( \(h \pm a, k\) ), where \(0

Step-by-Step Solution

Answer

\(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{c^2-a^2} = 1\).

Step 1: Understand Hyperbola Orientation

The given foci are \((h \pm c, k)\) and the vertices are \((h \pm a, k)\). This indicates that the hyperbola is oriented horizontally since both pairs of points vary in the x-direction while sharing the same y-coordinate \(k\).

Step 2: Define the Standard Equation

For a horizontally oriented hyperbola, the standard equation is \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), where \(a\) is the distance from the center to each vertex along the x-axis, and \(b\) relates to the distance from the center vertically, determined by \(c^2 = a^2 + b^2\).

Step 3: Identify Hyperbola Parameters

We are given that the center of the hyperbola is \((h, k)\), the distance \(a\) is from the vertices \((h \pm a, k)\), and the foci are \(c\) units from the center along the x-axis \((h \pm c, k)\).

Step 4: Calculate b using the Relationship

Since \(c^2 = a^2 + b^2\), solve for \(b^2\): \(b^2 = c^2 - a^2\). Use the given relationship to find specific values if \(c\) and \(a\) are known, otherwise express as a formula.

Step 5: Write the Hyperbola Equation

With \(b^2 = c^2 - a^2\), substitute into the standard form: \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{c^2-a^2} = 1\]. This represents the equation of the hyperbola given the conditions.

Key Concepts

Foci

The foci of a hyperbola are two crucial points that help in defining its shape. For a hyperbola centered at point \(h, k\), with a horizontal orientation, the foci are located at \(h \pm c, k\).
These points are key in understanding the hyperbola's eccentricity and measure how stretched it is compared to a circle. No matter where a point on the hyperbola is located, the difference in distances from this point to the two foci remains constant, and this characteristic is what primarily defines a hyperbola.

• The foci lie on the transverse axis.
• The distance between the center and each focus is denoted as \(c\).

Understanding the foci's position and role is essential to studying the properties of hyperbolas and their behavior in coordinate fashion.

Vertices

In a hyperbola, the vertices are the points where the hyperbola intersects its transverse axis. If the hyperbola is centered at \(h, k\) and oriented horizontally, its vertices are located at \(h \pm a, k\).
The distance between the center and each vertex is denoted by \(a\), reflecting the semi-major axis length of this conic section. Vertices form the closest points on the hyperbola to the center.

• The transverse axis is the line that joins the vertices.
• The vertices give the initial direction of the curve spread.

Vertices are fundamental in determining the basic shape of a hyperbola and offer insight into the hyperbola's overall dimensions.

Equation of Hyperbola

The equation of a hyperbola provides a mathematical relationship that describes every point on the hyperbolic curve. For a horizontally oriented hyperbola centered at \(h, k\), the equation typically is:\[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]
Here, \(a\) is the semi-major axis, and \(b\) relates to the distance from the center to the asymptotes along the transverse axis.
This equation encapsulates the entire set of hyperbola parameters:

• \(h\) and \(k\) are center coordinates.
• \(a\) is derived from the vertices position.
• \(b\) is calculated using the relation \(b^2 = c^2 - a^2\).

With this equation, you can graph the hyperbola and analyze its distinct properties.

Standard Form

The standard form of a hyperbola's equation simplifies its representation and understanding. The standard form for a horizontally oriented hyperbola centered at \(h, k\) is:\[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]
This representation allows one to quickly identify the center, vertices, and passage into alternate forms for varied applications or geometric interpretations.
Unlike ellipses and circles, a hyperbola's equation contains a subtraction between two squared terms, highlighting its distinct shape.

• The positive term relates to the transverse axis.
• The negative term corresponds to the conjugate axis direction.

The standard form is invaluable in mathematical analyses, offering a unified format to compare different hyperbolas efficiently.

Conic Sections

Hyperbolas belong to the overarching family of conic sections, which also includes circles, ellipses, and parabolas. These shapes are derived from slicing a double-napped cone at various angles.
A hyperbola is the result when the plane cuts through both nappes, forming two separate curves.

• Hyperbolas are distinguished by their open-ended curves.
• Their distinct property is defined by the constant difference in distances from any point on the curve to the foci.

Understanding hyperbolas in the context of conic sections helps visualize their geometric properties and their relations to other mathematical structures. Conic sections are vital in fields ranging from astronomy to architecture due to their diverse properties.