In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x^2/9−y^2/25=1

Verified step by step guidance

1. Identify the standard form of the hyperbola equation: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Here, \( a^2 = 9 \) and \( b^2 = 25 \).
2. Calculate \( a \) and \( b \) by taking the square roots: \( a = 3 \) and \( b = 5 \).
3. Determine the vertices of the hyperbola. Since the hyperbola is horizontal, the vertices are at \( (\pm a, 0) = (\pm 3, 0) \).
4. Find the foci using the formula \( c^2 = a^2 + b^2 \). Calculate \( c \) and locate the foci at \( (\pm c, 0) \).
5. Write the equations of the asymptotes. For a horizontal hyperbola, the asymptotes are \( y = \pm \frac{b}{a}x \), which simplifies to \( y = \pm \frac{5}{3}x \).

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, where (h, k) is the center, and a and b determine the distance to the vertices and co-vertices.

Asymptotes of a Hyperbola

Asymptotes are lines that the branches of a hyperbola approach but never touch. For a hyperbola in the 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 framework for sketching the hyperbola and indicate its direction and spread.

Foci of a Hyperbola

The foci of a hyperbola are two fixed points located along the transverse axis, which help define the shape of the hyperbola. For a hyperbola in standard form, the distance from the center to each focus is given by c = √(a² + b²). The foci are crucial for understanding the hyperbola's properties, including its eccentricity, which measures how 'stretched' the hyperbola is.