Conjugate Hyperbola: Find Equation w/ Asymptotes y=+/-2x

In summary, a conjugate hyperbola is a type of hyperbola with identical asymptotes. The equation for one pair of conjugate hyperbolas with asymptotes y = +/- 2x is given by \frac{x^2}{1} - \frac{y^2}{4} = 1 and \frac{x^2}{1} - \frac{y^2}{4} = -1. These equations can also be written as \frac{y^2}{4} - \frac{x^2}{1} = 1. One graph can be used to sketch both hyperbolas since they share the same asymptotes.
  • #1
blue_soda025
26
0
What is a conjugate hyperbola? I'm asked to find the equation of the conjugate hyperbola if the asymptotes are y = +/- 2x.
Would it be [tex]\frac{x^2}{1} + \frac{y^2}{4} = 1[/tex] or [tex]\frac{x^2}{1} + \frac{y^2}{4} = -1[/tex]?
 
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  • #2
blue_soda025 said:
What is a conjugate hyperbola? I'm asked to find the equation of the conjugate hyperbola if the asymptotes are y = +/- 2x.
Would it be [tex]\frac{x^2}{1} + \frac{y^2}{4} = 1[/tex] or [tex]\frac{x^2}{1} + \frac{y^2}{4} = -1[/tex]?
You forgot the all important (-) signs! Conjugate hyperbolas have identical asymptotes. One pair of conjugate hyperbolas having the above asymptotes is given by Eq #1 & #2:

[tex] :(1): \ \ \ \ \frac{x^2}{1} - \frac{y^2}{4} = 1[/tex]

[tex] :(2): \ \ \ \ \frac{x^2}{1} - \frac{y^2}{4} = -1 \ \ Or \ Equivalently \ \ \frac{y^2}{4} - \frac{x^2}{1} = 1 [/tex]



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  • #3
Oops, what was I thinking when I wrote that.

So I guess I sketch two graphs for this question.
 
  • #4
blue_soda025 said:
Oops, what was I thinking when I wrote that.

So I guess I sketch two graphs for this question.
One (1) graph should suffice. Both conjugate hyperbolas fit nicely on 1 graph since 1 hyperbola will graph above-&-below the asymptotes and the other left-&-right. (They both share the same asymptotes.)


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1. What is a conjugate hyperbola?

A conjugate hyperbola is a type of hyperbola that has two branches that are symmetrical about both the x-axis and the y-axis. It is also known as an equilateral hyperbola, as the distance from the center to each branch is equal.

2. How do you find the equation of a conjugate hyperbola with asymptotes y=+/-2x?

To find the equation of a conjugate hyperbola with asymptotes y=+/-2x, you can use the standard form of a hyperbola: (x-h)^2 / a^2 - (y-k)^2 / b^2 = 1. The center of the hyperbola will be at the point (h,k), and the values of a and b can be determined by the distance from the center to the vertices and the foci, respectively. In this case, since the asymptotes are y=+/-2x, the value of b will be equal to 2a, and the values of a and h can be found by substituting the coordinates of the center into the equation of the asymptotes.

3. What are the properties of a conjugate hyperbola?

A conjugate hyperbola has several important properties, including:

  • Two branches that are symmetrical about both the x-axis and the y-axis
  • Asymptotes with equations y=+/-a/b*x, where a and b are the lengths of the semi-major and semi-minor axes, respectively
  • Vertices at (h+a,k) and (h-a,k), where (h,k) is the center of the hyperbola
  • Foci at (h+sqrt(a^2+b^2),k) and (h-sqrt(a^2+b^2),k)
  • Eccentricity of the hyperbola, which is equal to c/a, where c is the distance from the center to the foci

4. How do you graph a conjugate hyperbola?

To graph a conjugate hyperbola, you can follow these steps:

  1. Determine the center of the hyperbola by finding the midpoint between the vertices
  2. Plot the vertices, foci, and center on a coordinate plane
  3. Sketch the asymptotes by drawing lines through the center that are parallel to the given equations
  4. Using the values of a and b, draw the hyperbola's branches, making sure they are symmetrical about the x-axis and y-axis
  5. Label the vertices, foci, and asymptotes on the graph

5. What are some real-life applications of conjugate hyperbolas?

Conjugate hyperbolas have many real-life applications, including:

  • Antennas and satellite dishes, which use the shape of a hyperbola to focus radio waves
  • Parabolic mirrors, which use the shape of a hyperbola to reflect light and create images
  • Optical instruments, such as telescopes and microscopes, which use hyperbolic lenses to manipulate light
  • Designing roller coasters, where hyperbolic shapes are used to create smooth and thrilling rides
  • Modeling the orbits of comets and other celestial objects

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