Chapter 10: Problem 46
Determine the length of the latus rectum (see Problem 45) of the hyperbola \(x^{2} / a^{2}-y^{2} / b^{2}=1\).
Short Answer
Expert verified
The length of the latus rectum is \( \frac{2b^2}{a} \).
Step by step solution
01
Identify the Standard Equation
The standard equation of a hyperbola with a horizontal transverse axis is given as \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Here, \(a\) and \(b\) are constants that determine the shape and dimensions of the hyperbola.
02
Understand Latus Rectum Definition
The latus rectum of a hyperbola refers to the line perpendicular to the transverse axis that passes through the foci of the hyperbola.
03
Formula for Latus Rectum Length
For a hyperbola of the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the length of the latus rectum is given by the formula \( \frac{2b^2}{a} \). This is derived from the properties of hyperbolas and their geometric definitions.
04
Plug Values into Formula
Insert the values from the standard form into the latus rectum formula. Here, the values for \(a\) and \(b\) come directly from the hyperbola's equation, and you find the latus rectum length as \( \frac{2b^2}{a} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Latus Rectum
The latus rectum of a hyperbola is a significant concept in understanding the shape and properties of the curve. It is defined as the line segment perpendicular to the transverse axis and passes through the focal points of the hyperbola. This means the latus rectum gives valuable information about how wide the hyperbola opens around its foci.
One practical aspect of the latus rectum is determining its length, which depends on the hyperbola's equation. For a hyperbola given by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the formula for the length of the latus rectum is \( \frac{2b^2}{a} \).
This formula is derived from the geometric properties of hyperbolas and the relationship between the semi-major and semi-minor axes. By inserting the specific values of \(a\) and \(b\) from the standard equation, we can quickly determine the length of the latus rectum.
One practical aspect of the latus rectum is determining its length, which depends on the hyperbola's equation. For a hyperbola given by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the formula for the length of the latus rectum is \( \frac{2b^2}{a} \).
This formula is derived from the geometric properties of hyperbolas and the relationship between the semi-major and semi-minor axes. By inserting the specific values of \(a\) and \(b\) from the standard equation, we can quickly determine the length of the latus rectum.
Standard Equation of a Hyperbola
When discussing hyperbolas, the standard equation forms the basis for understanding their structure. A hyperbola with a horizontally aligned transverse axis is represented by the standard equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).
Here, the terms \(a^2\) and \(b^2\) play crucial roles. The value of \(a\) determines the extent of the transverse axis, while \(b\) relates to the conjugate axis, which is perpendicular to the transverse axis. These terms essentially dictate the spacing of the vertices and foci along the axis.
Recognizing this standard form helps in analyzing and solving various problems related to hyperbolas, especially when different transformations or problem conditions are applied.
Here, the terms \(a^2\) and \(b^2\) play crucial roles. The value of \(a\) determines the extent of the transverse axis, while \(b\) relates to the conjugate axis, which is perpendicular to the transverse axis. These terms essentially dictate the spacing of the vertices and foci along the axis.
Recognizing this standard form helps in analyzing and solving various problems related to hyperbolas, especially when different transformations or problem conditions are applied.
Transverse Axis
The transverse axis of a hyperbola is an essential concept for understanding its shape and orientation. It is the line that passes through the two foci of the hyperbola and forms the longest distance between its vertices.
In the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the transverse axis is horizontal, meaning it runs along the x-axis. This axis plays a vital role because it sets the direction in which the hyperbola opens.
The length of the transverse axis is determined by \(2a\), which is the full distance between the vertices along the axis. Understanding this helps not only in graphing the hyperbola but also in solving various mathematical problems related to the hyperbola's properties.
In the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the transverse axis is horizontal, meaning it runs along the x-axis. This axis plays a vital role because it sets the direction in which the hyperbola opens.
The length of the transverse axis is determined by \(2a\), which is the full distance between the vertices along the axis. Understanding this helps not only in graphing the hyperbola but also in solving various mathematical problems related to the hyperbola's properties.
Geometric Definitions of Hyperbolas
Hyperbolas are unique curves defined by two central points, known as foci. A hyperbola consists of two separate branches, each bending away from the opposite focal point.
The fundamental geometric definition is that for any point on the hyperbola, the absolute difference in distances to the two foci is constant. This property distinguishes hyperbolas from other conic sections and gives them their unique shape.
Understanding this geometric principle is foundational in analyzing hyperbolas, especially when exploring their standard equations, axes, and determining aspects like the latus rectum or identifying foci. This geometric basis enables a deeper comprehension of hyperbolas in both theoretical and practical contexts.
The fundamental geometric definition is that for any point on the hyperbola, the absolute difference in distances to the two foci is constant. This property distinguishes hyperbolas from other conic sections and gives them their unique shape.
Understanding this geometric principle is foundational in analyzing hyperbolas, especially when exploring their standard equations, axes, and determining aspects like the latus rectum or identifying foci. This geometric basis enables a deeper comprehension of hyperbolas in both theoretical and practical contexts.
