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Chapter 9: Problem 40

Determine the equation in standard form of the hyperbola that satisfies the given conditions. Foci at (5,0),(-5,0)\(;\) passes through the point (3,0)

Short Answer

Expert verified
The standard form equation of the hyperbola given the conditions is \(x^2/9 - y^2/16 = 1\).

Step by step solution

01

Determining the Value of 'a'

The distance from the center to a vertex is denoted as 'a'. Since the hyperbola passes through the point (3,0), 'a' is equal to the x-coordinate of this point. Therefore, 'a' is 3.
02

Determining the Value of 'c'

The distance from the center to a focus, which is referred to as 'c', can be calculated as the distance from the center to one of the foci. Since one the foci is given at (5,0), 'c' is equal to the x-coordinate of this focus. Therefore, 'c' is 5.
03

Determining the Value of 'b'

Given that 'a' is 3 and 'c' is 5, we substitute these values into the following equation to find the value of 'b': \(c=\sqrt{a^2+b^2}\). Squaring both sides and rearranging gives \(b=\sqrt{c^2-a^2}\). Substituting our known values, \( b=\sqrt{5^2-3^2}=4\)
04

Forming the Standard Form Equation of the Hyperbola

Using the standard form equation \(x^2/a^2 - y^2/b^2 = 1\) and substituting for 'a' as 3 and 'b' as 4, the equation becomes \(x^2/3^2 - y^2/4^2 = 1\). Upon simplifying, we obtain the hyperbola equation as \(x^2/9 - y^2/16 = 1\).

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Key Concepts

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

Standard Form
The standard form of a hyperbola provides a structured way to represent its equation. A hyperbola, similar to an ellipse, has its own unique standard form that reflects its geometric properties. In the scenario where the transverse axis is along the x-axis, the standard form of the equation is:

x/y Hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)

In this format:
  • \( a \) is the distance from the center to each vertex along the x-axis.
  • \( b \) is related to the distance along the y-axis and can be evaluated using the relationship between the focal distance.
  • The number 1 on the right side of the equation indicates that the hyperbola has been "normalized" to simplify calculations and comparisons.
The center of the hyperbola is located at the origin (0,0) since there are no additional terms affecting \( x \) or \( y \).

Visualizing how the standard form is constructed helps in plotting the hyperbola and predicting its behavior.
Foci
The foci of a hyperbola are two fixed points located on the transverse axis of the hyperbola. They are significant because all points on the hyperbola maintain a constant difference in distances to these two points.

In the given problem:
  • The foci are at points (5,0) and (-5,0), which are symmetric around the origin.
  • The coordinate form of the foci helps in understanding the stretch and shape of the hyperbola.
  • The formula \( c = \sqrt{a^2 + b^2} \) is utilized to find the relationship between the foci and vertices.
For the given foci positions:- \( c = 5 \), as indicated by the x-coordinate of the foci.

The focus placement is vital in shaping the hyperbola's curve and ensuring the definition of a hyperbola is met, with the constant difference in distances being preserved.
Equation of a Hyperbola
Every hyperbola can be represented by a specific equation derived from its geometric features. Here, we will determine the equation based on provided conditions.

To find the equation of a hyperbola given the foci (at \((5,0)\) and \((-5,0)\)) and a point \((3,0)\) it passes through:
  • First, determine \( a \) as the distance from the center to a vertex. With the point (3,0), \( a \) equals 3.
  • Next, \( c \), which measures the distance from the center to a focus, is 5.
  • We found \( b \) using \( b = \sqrt{c^2 - a^2} \), resulting in \( b = 4 \).
Finally, using the standard form:\

\( \frac{x^2}{9} - \frac{y^2}{16} = 1 \).

This formula succinctly represents the hyperbola's shape and orientation, encapsulating all vital elements such as foci, vertices, and center. It makes calculation and analysis straightforward by providing clear relationships between all key parameters.

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Most popular questions from this chapter

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