Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci: (0,-4), (0,4); Vertices: (0, -2), (0,2)

Verified step by step guidance

1. Identify the orientation of the hyperbola. Since the foci and vertices are aligned along the y-axis (x-coordinates are the same), the hyperbola is vertical. The standard form of a vertical hyperbola is: $\frac{y^2}{a^2}-\frac{x^2}{b^2}=\; 1$.
2. Determine the center of the hyperbola. The center is the midpoint of the vertices. Since the vertices are (0, -2) and (0, 2), the center is at (0, 0).
3. Find the value of $a^2$. The distance from the center to each vertex is $a$. Here, the distance is 2, so $a^2=\; 4$.
4. Find the value of $c^2$. The distance from the center to each focus is $c$. Here, the distance is 4, so $c^2=\; 16$.
5. Use the relationship $c^2=a^2+b^2$ to find $b^2$. Substituting $c^2=\; 16$ and $a^2=\; 4$, solve for $b^2$. Once you have $b^2$, substitute the values of $a^2$ and $b^2$ into the standard form equation.

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 depends on its orientation, which can be horizontal or vertical, determined by the positions of its foci and vertices.

Foci and Vertices

In a hyperbola, the foci are two fixed points located along the transverse axis, which is the line segment that connects the vertices. The vertices are the points where the hyperbola intersects its transverse axis. The distance between the center and each vertex is denoted as 'a', while the distance from the center to each focus is 'c'. The relationship between 'a', 'b' (the distance to the co-vertices), and 'c' is given by the equation c² = a² + b².

Standard Form of a Hyperbola

The standard form of a hyperbola's equation is expressed as (y²/a²) - (x²/b²) = 1 for a vertical hyperbola, and (x²/a²) - (y²/b²) = 1 for a horizontal hyperbola. In this case, since the foci and vertices are aligned vertically, the equation will take the vertical form. The values of 'a' and 'c' can be derived from the given vertices and foci, allowing for the complete equation to be formulated.