Problem 67 - College Algebra

Question

What is a hyperbola?

Step-by-Step Solution

Answer

A hyperbola is a type of curve in a plane formed by the set of all points wherein the absolute difference of the distances from two fixed points called foci is constant. It has components like a center, foci, vertices, and directrices. Hyperbola can be oriented either horizontally or vertically and have standard equations accordingly.

Step 1: Definition of Hyperbola

A hyperbola is a type of curve in a plane. The simple definition is that it is the set of all points \(P\) in the plane such that the absolute difference of the distances between \(P\) and two fixed points called foci, is a constant.

Step 2: Components of a Hyperbola

The components of a hyperbola include the following: \n- Center: This is the midpoint between the two foci.\n- Foci: These are the fixed points that define a hyperbola.\n- Vertices: These are the points of intersection of the hyperbola and a line through the foci.\n- Directrices: These are the lines perpendicular to the line through the foci which intersects the center.

Step 3: Standard Equations of Hyperbola

The standard equations of a hyperbola depend on whether it is oriented horizontally or vertically. \n- For a horizontal hyperbola, the standard equation is \((x - h)^2 / a^2 - (y - k)^2 / b^2 = 1\), where \((h, k)\) is the center and \(a^2\), \(b^2\) are the squares of the semi-major and semi-minor axes. \n- For a vertical hyperbola, the standard equation is \((y - k)^2 / a^2 - (x - h)^2 / b^2 = 1\), where \((h, k)\) is the center and \(a^2\), \(b^2\) are the squares of the semi-major and semi-minor axes.