In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 9x^2−4y^2=36

Verified step by step guidance

1. Rewrite the given equation in standard form by dividing all terms by 36: \( \frac{9x^2}{36} - \frac{4y^2}{36} = 1 \).
2. Simplify the equation: \( \frac{x^2}{4} - \frac{y^2}{9} = 1 \). This is the standard form of a hyperbola centered at the origin with a horizontal transverse axis.
3. Identify the values of \(a^2\) and \(b^2\): \(a^2 = 4\) and \(b^2 = 9\). Thus, \(a = 2\) and \(b = 3\).
4. Determine the vertices of the hyperbola. Since the transverse axis is horizontal, the vertices are at \((\pm a, 0)\), which are \((\pm 2, 0)\).
5. Find the equations of the asymptotes. For a hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the asymptotes are given by \(y = \pm \frac{b}{a}x\). Substitute \(a = 2\) and \(b = 3\) to get the equations \(y = \pm \frac{3}{2}x\).

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 or (y-k)²/a² - (x-h)²/b² = 1, where (h, k) is the center, and 'a' and 'b' determine the distances 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 standard form, the equations of the asymptotes can be derived from the center and the values of 'a' and 'b'. They are given by the equations y = k ± (b/a)(x - h) for horizontal hyperbolas and x = h ± (a/b)(y - k) for vertical hyperbolas, providing a guide for sketching the hyperbola.

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. The distance from the center to each focus is denoted by 'c', where c² = a² + b². The foci are crucial for understanding the hyperbola's properties, as they determine the distance relationship that characterizes hyperbolas, specifically that the difference in distances from any point on the hyperbola to the two foci is constant.