Find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (−4, 0), (4, 0); vertices:(−3, 0), (3, 0)

Verified step by step guidance

1. Identify the center of the hyperbola by finding the midpoint of the vertices. Since the vertices are at (−3, 0) and (3, 0), the center is at the midpoint: \(\left( \frac{-3 + 3}{2}, \frac{0 + 0}{2} \right)\).
2. Determine the orientation of the hyperbola. Because the vertices and foci lie on the x-axis, the transverse axis is horizontal, so the standard form will be \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\) where \((h, k)\) is the center.
3. Calculate the distance \(a\), which is the distance from the center to each vertex. Since the vertices are at (−3, 0) and (3, 0), \(a\) is the distance from the center to either vertex along the x-axis.
4. Calculate the distance \(c\), which is the distance from the center to each focus. The foci are at (−4, 0) and (4, 0), so \(c\) is the distance from the center to either focus along the x-axis.
5. Use the relationship for hyperbolas: \(c^2 = a^2 + b^2\). Solve for \(b^2\) using the values of \(a\) and \(c\) found, then write the standard form equation as \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\).

Key Concepts

Standard Form of a Hyperbola

The standard form of a hyperbola equation depends on its orientation. For a hyperbola centered at the origin with foci and vertices on the x-axis, the equation is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Here, \(a\) is the distance from the center to each vertex, and \(b\) relates to the conjugate axis.

Relationship Between Vertices, Foci, and Parameters a, b, c

In a hyperbola, \(a\) is the distance from the center to each vertex, and \(c\) is the distance from the center to each focus. These satisfy the equation \(c^2 = a^2 + b^2\). Knowing the coordinates of vertices and foci allows calculation of \(a\), \(c\), and subsequently \(b\).

Center and Orientation of the Hyperbola

The center of the hyperbola is the midpoint between the vertices and foci. Since the given points lie on the x-axis, the hyperbola opens left and right. This orientation determines the form of the equation and which variable is positive or negative in the standard form.