Problem 58 - Precalculus: Functions and Graphs

Question

Find an equation for the indicated part of the hyperbola. Upper branch of \(\frac{y^{2}}{49}-\frac{x^{2}}{25}=1\)

Step-by-Step Solution

Answer

The equation for the upper branch is \(y = \sqrt{49\left(1 + \frac{x^{2}}{25}\right)}\) with \(y \geq 0\).

Step 1: Identify the Given Equation

We are given the equation of a hyperbola: \(\frac{y^{2}}{49} - \frac{x^{2}}{25} = 1\). This is in the standard form for a vertical hyperbola.

Step 2: Determine the Vertices

For a hyperbola in the form \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\), the vertices are \((0, \pm a)\). Given \(a^{2} = 49\), we have \(a = 7\). Thus, the vertices are \((0, 7)\) and \((0, -7)\).

Step 3: Equation for the Upper Branch

The upper branch of the hyperbola is above the x-axis, so it corresponds to the vertex \((0, 7)\). The equation remains \(\frac{y^{2}}{49} - \frac{x^{2}}{25} = 1\), but represents points where \(y \geq 0\).

Step 4: Express the Upper Branch Condition

To specifically express the upper branch, resolve the equation \(y = \sqrt{49\left(1 + \frac{x^{2}}{25}\right)}\) to only consider the positive root for \(y\), resulting in the equation for the upper branch.

Key Concepts

Conic Sections

Conic sections are among the fascinating subjects of geometry. They are the curves obtained by intersecting a plane with a double-napped cone. The main types of conic sections are:

• Circle
• Ellipse
• Parabola
• Hyperbola

These shapes appear in numerous natural and human-made forms. Each has unique properties and equations that define their shape. For instance, a hyperbola is composed of two separate curves called branches.
It's essential to understand that the nature of a conic section depends on the angle and location of the intersecting plane. Understanding conics helps in various fields like astronomy, physics, and engineering. In this exercise, the focus is on hyperbolas, which appear when the plane cuts both halves of a double cone and does not pass through the apex.

Vertical Hyperbola

A vertical hyperbola is a type of hyperbola where its transverse axis is vertical. The standard equation for a vertical hyperbola is \[\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\]This indicates that the hyperbola opens upwards and downwards.
The terms in the equation are crucial:

• \(a^2\) is associated with the term \(y^2\), determining the distance from the center to the vertices along the vertical axis.
• \(b^2\) is tied to the \(x^2\) term, related to the distance to the asymptotes' slopes.

Vertical hyperbolas are unique because both their branches extend in the vertical direction, sharply diverging as they move away from the center. In the given exercise, the equation \[\frac{y^{2}}{49} - \frac{x^{2}}{25} = 1\]represents a vertical hyperbola with its center at the origin and opening upwards and downwards.

Vertices of Hyperbola

The vertices of a hyperbola are the points where each branch is closest to the center. For a hyperbola in the form \[\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1,\]the vertices are determined by the value of \(a\).
Here’s how you find them:

• Calculate \(a\) from the equation, which is the square root of \(a^2\). For example, if \(a^2 = 49\), then \(a = 7\).
• The vertices for a vertical hyperbola are located at \((0, \pm a)\). Thus, in our case, they are at \((0, 7)\) and \((0, -7)\).

These vertices serve as important reference points. They determine the shape of the hyperbola and the direction it opens. By understanding where the vertices are, we gain insight into the hyperbola's structure and can sketch its graph accurately.