In Exercises 5–12, find the standard form of the equation of each hyperbola satisfying the given conditions. Endpoints of transverse axis: (0, −6), (0, 6); asymptote: y=2x

Verified step by step guidance

1. Identify the center of the hyperbola. Since the endpoints of the transverse axis are (0, -6) and (0, 6), the center is the midpoint of these points, which is (0, 0).
2. Determine the orientation of the hyperbola. The transverse axis is vertical because the x-coordinates of the endpoints are the same. Therefore, the equation of the hyperbola will have the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).
3. Calculate the length of the transverse axis. The distance between the endpoints (0, -6) and (0, 6) is 12, so the length of the transverse axis is 12, and \( 2a = 12 \), giving \( a = 6 \).
4. Use the given asymptote equation \( y = 2x \) to find \( b \). For a vertical hyperbola, the slopes of the asymptotes are \( \pm \frac{a}{b} \). Since \( \frac{a}{b} = 2 \) and \( a = 6 \), solve for \( b \) to get \( b = 3 \).
5. Write the standard form of the equation of the hyperbola using the values of \( a \) and \( b \). The equation is \( \frac{y^2}{36} - \frac{x^2}{9} = 1 \).

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's equation can be expressed as (y-k)²/a² - (x-h)²/b² = 1 for vertical hyperbolas, where (h, k) is the center, and 'a' and 'b' are distances related to the transverse and conjugate axes.

Transverse Axis

The transverse axis of a hyperbola is the line segment that connects the two vertices of the hyperbola. In this case, the endpoints of the transverse axis are given as (0, -6) and (0, 6), indicating that the hyperbola opens vertically. The length of the transverse axis is determined by the distance between these two points, which is essential for finding the values of 'a' in the hyperbola's equation.

Asymptotes

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'. Given the asymptote equation y = 2x, we can determine the slope and use it to find the relationship between 'a' and 'b', which is crucial for writing the hyperbola's equation in standard form.