Problem 67 - Algebra and Trigonometry

Question

What is a hyperbola?

Step-by-Step Solution

Answer

A hyperbola is a type of conic section defined by a specific equation and is the set of all points in a plane such that the difference of the distances from two fixed points is constant. It has two disconnected curves that open either right and left or up and down.

Step 1: Definition

A hyperbola is a type of conic section, defined by its specific equation. It is the set of all points in a plane such that the difference of the distances from two fixed points (the foci) is constant for all points.

Step 2: Description of Shape

A hyperbola is a mirror image of itself across a central line, called the axis of the hyperbola. It is composed of two disconnected curves that open either up and down or right and left.

Step 3: Equation

The standard form equation of a hyperbola is given by \( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \) if the hyperbola is oriented horizontally, or \( \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \) if it is oriented vertically, where (h, k) is the center of the hyperbola, and a and b are constants that determine the shape of the hyperbola.