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Chapter 10: Problem 116

For the following exercises, given information about the graph of the hyperbola, find its equation. Center: \((0,0) ;\) vertex: \((0,-13) ;\) one focus: \((0, \sqrt{313})\)

Short Answer

Expert verified
The equation of the hyperbola is \( \frac{y^2}{169} - \frac{x^2}{144} = 1 \).

Step by step solution

01

Identify Hyperbola Type

Since the vertex and focus are both aligned along the y-axis, the hyperbola is vertical. The general equation for a vertical hyperbola centered at the origin is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).
02

Find \(a^2\) using the Vertex

The vertex form of a vertical hyperbola is \((0, \pm a)\). Given the vertex \((0, -13)\), we have \(a = 13\). Therefore, \(a^2 = 13^2 = 169\).
03

Calculate \(c^2\) using the Focus

The focus form for a vertical hyperbola is \((0, \pm c)\). Given a focus at \((0, \sqrt{313})\), we have \(c = \sqrt{313}\), so \(c^2 = 313\).
04

Find \(b^2\) using the Hyperbola Relationship

For hyperbolas, it is true that \(c^2 = a^2 + b^2\). Substituting the known values, \(313 = 169 + b^2\). Solving for \(b^2\), we find \(b^2 = 313 - 169 = 144\).
05

Write the Equation of the Hyperbola

Substitute \(a^2\) and \(b^2\) into the standard vertical hyperbola equation: \( \frac{y^2}{169} - \frac{x^2}{144} = 1 \). Therefore, the equation of the hyperbola is \( \frac{y^2}{169} - \frac{x^2}{144} = 1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Vertical Hyperbolas
A hyperbola is a type of conic section that consists of two disconnected curves, or "branches". In the case of a vertical hyperbola, the branches open upwards and downwards, along the y-axis. The orientation is crucial: it tells us how the branches are aligned. For a vertical hyperbola, the standard equation is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). The key features are the following:

  • The center of the hyperbola is a fixed point, typically denoted as \((h, k)\). In simpler cases like this one, it is at the origin \((0,0)\).
  • The vertices are directly above and below the center if the hyperbola is vertical.
  • The terms \(a\) and \(b\) represent distances from the center, influencing the shape and spread of the hyperbola.
Knowing it's vertical helps set up the equation properly, guiding us on which axis the components align along.
Locating the Focus of a Hyperbola
The foci, or focus points, of a hyperbola are significant because they determine the curvature of its branches. For a vertical hyperbola, the foci also lie along the y-axis. Here's how it works:

  • The coordinates of the foci are \((h, k \pm c)\) if the center is at \((h, k)\). Typically, in a simple hyperbola centered at the origin, these are just \((0, \pm c)\).
  • The distance \(c\) is calculated based on the distances \(a\) and \(b\), where \(c^2 = a^2 + b^2\).
  • This means that the foci are located a distance \(c\) from the center along the axis of the hyperbola (y-axis in this case).
The foci help us understand how wide the hyperbola opens and are crucial for accurately forming the equation of the curve.
Defining the Vertex of a Hyperbola
The vertex represents the "turning point" where each branch of the hyperbola changes direction. For a vertical hyperbola, this understanding is especially handy:

  • The vertices of a vertical hyperbola are situated at \((h, k \pm a)\). Here, with a center at \((0,0)\), they are \((0, \pm a)\).
  • The distance \(a\) is the distance from the center to a vertex along the line of symmetry (again, the y-axis).
  • The given vertex \((0, -13)\) signifies that \(a = 13\), hence providing valuable information for formulating the hyperbola's equation.
The vertex provides a clear directional guide for the hyperbola's branches and is fundamental in establishing the formulaic structure of the graph.
Understanding the Hyperbola Standard Form
The standard form of a hyperbola's equation provides a concise representation of its important features. For a vertical hyperbola, it's written as \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). Here's a breakdown:

  • The numerator \(y^2\) tells us the branches are opening along the y-axis, confirming the hyperbola's vertical orientation.
  • \(a^2\) and \(b^2\) are pivotal as they denote squared distances from the center to the vertices and co-vertices, respectively. Calculating these involves understanding the vertex and focus positions.
  • The equation \(c^2 = a^2 + b^2\) links all these components, allowing us to calculate \(b^2\) if we know \(a^2\) and \(c^2\).
By substituting these variables into the standard equation, one can accurately describe the shape and position of the hyperbola within a coordinate plane.

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Most popular questions from this chapter

For the following exercises, given information about the graph of the hyperbola, find its equation. Center: \((4,2) ;\) vertex: \((9,2) ;\) one focus: \((4+\sqrt{26}, 2)\)

Write the equation of the ellipse in standard form. Then identify the center, vertices, and foci. $$\frac{(x-2)^{2}}{100}+\frac{(y+3)^{2}}{36}=1$$

Graph the ellipse, noting center, vertices, and foci. $$\frac{(x-4)^{2}}{25}+\frac{(y+3)^{2}}{49}=1$$

Graph the hyperbola, labeling vertices and foci. $$x^{2}-4 y^{2}+6 x+32 y-91=0$$

Write the equation of the hyperbola in standard form. Then give the center, vertices, and foci. $$9 y^{2}-4 x^{2}+54 y-16 x+29=0$$
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