Find the standard form of the equation of each hyperbola satisfying the given conditions. Center: (4, −2); Focus: (7, −2); vertex: (6, −2)

Verified step by step guidance

1. Identify the center of the hyperbola as given: \( (h, k) = (4, -2) \).
2. Determine the orientation of the hyperbola by comparing the coordinates of the focus and vertex with the center. Since the y-coordinates are the same (\(-2\)) and the x-coordinates differ, the hyperbola opens horizontally.
3. Calculate the distance between the center and a vertex to find \(a\): \(a = |6 - 4| = 2\).
4. Calculate the distance between the center and a focus to find \(c\): \(c = |7 - 4| = 3\).
5. Use the relationship \(c^2 = a^2 + b^2\) to solve for \(b^2\): \(b^2 = c^2 - a^2 = 3^2 - 2^2\). Then write the standard form of the hyperbola with a horizontal transverse axis: \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\).

Key Concepts

Standard Form of a Hyperbola

The standard form of a hyperbola's equation depends on its orientation and center. For a hyperbola centered at (h, k), if it opens horizontally, the form is ((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1; if it opens vertically, the form is ((y - k)^2 / a^2) - ((x - h)^2 / b^2) = 1. Identifying the correct form is essential to write the equation properly.

Relationship Between Center, Vertex, and Focus

The center of a hyperbola is the midpoint between its vertices and foci. The distance from the center to a vertex is 'a', and the distance from the center to a focus is 'c'. These distances satisfy the relationship c^2 = a^2 + b^2, which helps determine the values needed for the equation.

Determining Orientation of the Hyperbola

The orientation of a hyperbola (horizontal or vertical) depends on the alignment of its vertices and foci. If the vertices and foci share the same y-coordinate, the hyperbola opens horizontally; if they share the same x-coordinate, it opens vertically. This orientation guides which variable is associated with 'a' and 'b' in the equation.