Graph the hyperbola. Locate the foci and find the equations of the asymptotes. ((x-2)^2)/25 - ((y+3)^2)/16 = 1

Verified step by step guidance

1. Step 1: Recognize the standard form of the hyperbola. The given equation is \( \frac{(x-2)^2}{25} - \frac{(y+3)^2}{16} = 1 \), which matches the standard form of a hyperbola that opens horizontally: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \). Here, \( h = 2 \), \( k = -3 \), \( a^2 = 25 \), and \( b^2 = 16 \).
2. Step 2: Identify the center of the hyperbola. The center is given by \( (h, k) \), so the center is \( (2, -3) \).
3. Step 3: Determine the vertices. For a horizontally opening hyperbola, the vertices are located \( a \) units to the left and right of the center along the x-axis. Since \( a = \sqrt{25} = 5 \), the vertices are \( (2-5, -3) = (-3, -3) \) and \( (2+5, -3) = (7, -3) \).
4. Step 4: Locate the foci. The distance from the center to each focus is \( c \), where \( c = \sqrt{a^2 + b^2} \). Calculate \( c = \sqrt{25 + 16} = \sqrt{41} \). The foci are \( (2-\sqrt{41}, -3) \) and \( (2+\sqrt{41}, -3) \).
5. Step 5: Find the equations of the asymptotes. For a horizontally opening hyperbola, the asymptotes are given by \( y - k = \pm \frac{b}{a}(x - h) \). Substituting \( b = \sqrt{16} = 4 \), \( a = 5 \), \( h = 2 \), and \( k = -3 \), the equations of the asymptotes are \( y + 3 = \frac{4}{5}(x - 2) \) and \( y + 3 = -\frac{4}{5}(x - 2) \).

Key Concepts

Hyperbola Definition

A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which are mirror images of each other. The standard form of a hyperbola's equation can be expressed as (x-h)²/a² - (y-k)²/b² = 1 for horizontal hyperbolas, where (h, k) is the center, and a and b determine the distances to the vertices and co-vertices.

Foci of a Hyperbola

The foci of a hyperbola are two fixed points located along the transverse axis, which is the line segment that connects the vertices of the hyperbola. The distance from the center to each focus is denoted as c, where c² = a² + b². The foci play a crucial role in defining the shape of the hyperbola and are used in various applications, including navigation and physics.

Asymptotes of a Hyperbola

Asymptotes are lines that the branches of a hyperbola approach but never touch. For a hyperbola in the standard form (x-h)²/a² - (y-k)²/b² = 1, the equations of the asymptotes can be derived as y - k = ±(b/a)(x - h). These lines provide a visual guide for the behavior of the hyperbola as it extends towards infinity, helping to sketch the graph accurately.