Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the given central conic. Hyperbola with a vertex at \((0,-4)\) and a focus at \((0,-5)\)

Step-by-Step Solution

Answer

The given data leads to a degenerate case with no valid equation under normal assumptions.

Step 1: Understand the structure of a hyperbola

A hyperbola with a vertical transverse axis (aligned with the y-axis) can be expressed by the equation \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\), where \((h, k)\) is the center. In this problem, both the vertex and the focus have x-coordinate 0, so the hyperbola is centered on the y-axis.

Step 2: Identify the center of the hyperbola

Since the vertex is at \((0, -4)\) and the focus is at \((0, -5)\), we can deduce that the center lies halfway between them. The point halfway between \(-4\) and \(-5\) is \(-\frac{9}{2}\) or \(-4.5\). So the center \((h, k)\) is \((0, -4.5)\).

Step 3: Calculate the distance from the center to the vertex (a)

The distance from the center \((0, -4.5)\) to the vertex \((0, -4)\) is the value of \(a\). This distance is \(0.5\). This means \(a = 0.5\).

Step 4: Calculate the distance from the center to the focus (c)

The distance from the center \((0, -4.5)\) to the focus \((0, -5)\) is the value of \(c\). This distance is also \(0.5\), so \(c = 0.5\).

Step 5: Apply the relationship between a, b, and c

In a hyperbola, the relationship \(c^2 = a^2 + b^2\) holds true. Given that \(a = 0.5\) and \(c = 0.5\), we can solve for \(b\):\[(0.5)^2 = (0.5)^2 + b^2 \]\[0.25 = 0.25 + b^2 \]\[0 = b^2\]Then \(b = 0\).

Step 6: Write the equation of the hyperbola

Since the value of \(b\) is zero, the hyperbola essentially collapses into a degenerate case. The vertex and the focus coincide with the center, which mathematically implies a contradiction or an ill-defined hyperbola under standard form. Thus, an equation cannot be formed with horizontal/vertical asymmetry parameters.

Key Concepts

Central Conic

Central conics form an essential category within conic sections. They refer to ellipses and hyperbolas, both of which have a center. This center is a point equidistant from various points like vertices or foci, specific to the type of conic. In the case of a hyperbola or an ellipse, we find this center by taking the average position between two prominent features, such as vertices or foci, along the axis of symmetry.

• The center of a central conic is a critical part for deriving its equation.
• Understanding how to find the center helps in determining other parameters, such as distances to vertices and foci.

Central conics share the property of having reflective symmetry, which plays a crucial role in graphing and understanding their behavior.

Ellipse and Hyperbola Characteristics

Ellipses and hyperbolas are both conic sections, but they exhibit distinct characteristics that set them apart. They can be defined based on their geometric properties and equations:

• An ellipse is characterized by its two foci and the fact that the sum of distances from any point on the ellipse to these foci is constant.
• A hyperbola, on the other hand, consists of two separate curves whose difference in distances to two foci is constant.

Despite these differences, both share a shape that revolves around a central axis and center. The understanding of these characteristics is vital as it helps in the derivation and plotting of their equations. Hyperbolas, in particular, will have differing approaches in plotting due to their open curves as opposed to the closed curves of ellipses.

Conic Sections

Conic sections are curves derived from the intersection of a plane and a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. The nature of the section is determined by the angle formed between the plane and the axis of the cone.
Conic sections can be expressed in terms of algebraic equations:

• Ellipses and circles have a positive combined constant term in their main equation.
• Hyperbolas, on the other hand, have a difference of squared terms that indicate their nature.

Understanding conic sections is vital in various fields, from astronomy to physics, as they model paths of celestial bodies, projectiles, and many real-world phenomena.
In these equations, the coefficients and terms dictate the specific type and orientation of the conic section.

Equation of a Hyperbola

The equation of a hyperbola is central to identifying its properties and graphically representing it. A hyperbola has an equation of the form:\[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\]for a vertical transverse axis, where \(h, k\) is the center.

• The values \(a\) and \(b\) represent distances to the vertices or the semi-transverse and semi-conjugate axes, respectively.
• In solving for the equation, ensure that you solve for \(b\) using the relationship \(c^2 = a^2 + b^2\).

In our exercise, we encountered a unique situation where \(b = 0\), indicating a degenerate case of the hyperbola, leading to the understanding that not all sets of points yield a non-degenerate hyperbola.
This highlights the importance of careful attention to the constraints and calculations involved in defining hyperbolas.