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Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. x² − y² = 100

User Pavel Bely
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Final answer:

The given equation x² - y² = 100 represents a hyperbola. The vertices are (-10, 0) and (10, 0), the foci are (-10, 0) and (10, 0), and the asymptotes have equations y = 0. The graph is symmetric with respect to the y-axis and the origin is the center.

Step-by-step explanation:

The given equation x² - y² = 100 represents a hyperbola. To find the vertices, foci, and asymptotes, we can rewrite the equation in standard form as (x/h)² - (y/k)² = 1, where (h, k) is the center of the hyperbola.

Comparing the equation to the standard form, we have (x/0)² - (y/0)² = 1. So the center of the hyperbola is (0, 0).

The vertices are located at (±a, 0) where a is the distance from the center to the vertices. In this case, a = √100 = 10. So the vertices are (-10, 0) and (10, 0).

The foci are located at (±c, 0) where c is the distance from the center to the foci. In this case, c = √(a² + b²) = √(10² + 0²) = 10. So the foci are (-10, 0) and (10, 0).

The asymptotes have equations y = ±(b/a)x, where b is the distance from the center to the asymptotes. In this case, b = 0. So the asymptotes have equations y = 0.

Sketching the graph, we have a hyperbola centered at the origin (0, 0) with vertices at (-10, 0) and (10, 0), and the foci also at (-10, 0) and (10, 0). The asymptotes are the x and y axes. The graph is symmetric with respect to the y-axis and the origin is the center.

User AQuirky
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