How Do You Determine the Center and Intercepts of a Hyperbola?

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SUMMARY

The discussion focuses on determining the center and intercepts of the hyperbola defined by the equation 9(x-2)² - 16(y-4)² = 144. The center of the hyperbola is located at the point (2, 4), which is in the first quadrant. The x-intercepts can be found by setting y to 0, resulting in the points (6, 0) and (-2, 0), while the y-intercepts are found by setting x to 0, yielding the points (0, 2) and (0, 6).

PREREQUISITES
  • Understanding of hyperbola equations and their standard forms
  • Knowledge of coordinate geometry, specifically quadrants and intercepts
  • Familiarity with algebraic manipulation to isolate variables
  • Basic skills in solving equations involving squares and square roots
NEXT STEPS
  • Study the properties of hyperbolas, including asymptotes and vertices
  • Learn how to graph hyperbolas using their standard equations
  • Explore transformations of hyperbolas through shifts and scaling
  • Investigate the relationship between hyperbolas and conic sections
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Students studying algebra and geometry, educators teaching conic sections, and anyone seeking to enhance their understanding of hyperbolas and their properties.

Hygelac
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Hey, I have a couple of _easy_ questions about hyperbolas, but its been a while since I have worked with them and am not able to look them up in my math book currently...if someone could just get me started in the right direction, I would really appreciate it :)

Given the equation of the hyperbola: 9(x-2)^2 - 16(y-4)^2 = 144

1. In which quadrant is its center?

2. What are its x/y intercepts?

Again, if you could just get me started, that would be great :biggrin:
 
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Since you haven't shown any attempt at solving the problem, I will just get you started. The generalized equation for a horizontal hyperbola is:
\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1
where (h,k) is the center, a is the horizontal distance from the center to each vertex, and b is the vertical distance from the center to a point on one of the asymptotes (x,y) such that x is the x-coordinate of a vertex.
 

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