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Analyzing Hyperbolas: Finding Center, Foci, and Asymptotes

January 07, 2025Science1895
Analyzing Hyperbolas: Finding Center, Foci, and Asymptotes Understandi

Analyzing Hyperbolas: Finding Center, Foci, and Asymptotes

Understanding the properties of hyperbolas is crucial in mathematics, especially in fields involving calculus, algebra, and geometry. This article focuses on analyzing a specific hyperbola equation and finding its center, vertices, foci, and asymptote equations. The hyperbola equation we will analyze is 9x2 - 9y2 - 15x - 21y - 20 0. We will systematically transform this equation into its standard form and identify its key properties.

Step-by-Step Analysis

Step 1: Rearrange the Equation

Let's start by rearranging the given equation:

9x2 - 9y2 - 15x - 21y - 20 0

Rearranging gives:

9x2 - 15x - 9y2 - 21y 20

Step 2: Completing the Square

Next, we will complete the square for both the x and y terms:

For x2 - 15x and y2 - 21y:

x2 - 15x (x - 15/2)2 - (15/2)2 (x - 15/2)2 - 225/4

y2 - 21y (y - 21/2)2 - (21/2)2 (y - 21/2)2 - 441/4

Substituting these back into the equation:

9[(x - 15/2)2 - 225/4] - 9[(y - 21/2)2 - 441/4] 20

Simplifying gives:

9(x - 15/2)2 - 2025/4 - 9(y - 21/2)2 3969/4 20

9(x - 15/2)2 - 9(y - 21/2)2 20 - 1944/4 20

9(x - 15/2)2 - 9(y - 21/2)2 860/4 - 400/4

9(x - 15/2)2 - 9(y - 21/2)2 20 - 20

9(x - 15/2)2 - 9(y - 21/2)2 48/36 - 14/36

9(x - 15/2)2 - 9(y - 21/2)2 14/36

(x - 5/6)2 / (14/9) - (y - 7/6)2 / (14/9) 1

Step 3: Identify the Center, Vertices, Foci, and Asymptotes

The equation can now be expressed as:

(x - h2) / a2 - (y - k2) / b2 1

where h -5/6, k 7/6, a2 14/9, b2 14/9.

Center: (-5/6, 7/6) Vertices: (-5/6 pm sqrt(14)/3, 7/6) Foci: (-5/6 pm sqrt(14)/3, 7/6) Asymptotes: y - 7/6 pm sqrt(14)/3 (x 5/6)

Conclusion

The analysis of the hyperbola equation 9x2 - 9y2 - 15x - 21y - 20 0 has revealed the center, vertices, foci, and asymptotes. This type of problem is fundamental in understanding hyperbolas and can be applied in various fields of mathematics. By following the steps outlined above, you can confidently solve similar problems and gain a deeper understanding of hyperbolas.