In Exercises 33–42, use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x−1)^2−(y−2)^2=3

Verified step by step guidance

1. Step 1: Identify the center of the hyperbola. The center is given by the values in the parentheses. In this case, the center is at (1, 2).
2. Step 2: Identify the values of a and b. The equation of a hyperbola is given by \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\). In this case, a^2 = 3 and b^2 = 1. Therefore, a = sqrt(3) and b = 1.
3. Step 3: Identify the vertices of the hyperbola. The vertices are located at a distance of 'a' units from the center along the x-axis. In this case, the vertices are at (1±sqrt(3), 2).
4. Step 4: Identify the foci of the hyperbola. The foci are located at a distance of 'c' units from the center, where c^2 = a^2 + b^2. In this case, c = sqrt(3+1) = 2, so the foci are at (1±2, 2).
5. Step 5: Find the equations of the asymptotes. The equations of the asymptotes for a hyperbola centered at (h, k) are given by y = k ± (b/a)(x - h). In this case, the equations of the asymptotes are y = 2 ± (1/sqrt(3))(x - 1).

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 can be expressed as (x-h)²/a² - (y-k)²/b² = 1 for horizontal hyperbolas, or (y-k)²/a² - (x-h)²/b² = 1 for vertical hyperbolas, where (h, k) is the center.

Asymptotes of a Hyperbola

Asymptotes are lines that the branches of a hyperbola approach but never touch. For a hyperbola in standard form, the equations of the asymptotes can be derived from the center and the values of 'a' and 'b'. Specifically, for a hyperbola centered at (h, k), the asymptotes are given by y - k = ±(b/a)(x - h), which helps in sketching the graph accurately.

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 by 'c', where c² = a² + b². The foci play a crucial role in defining the shape of the hyperbola and are essential for understanding its geometric properties.