Question

Give the property that defines all hyperbolas.

Short answer

Answer: The property that defines all hyperbolas is that the difference of distances from two fixed points, called foci, to any point on the curve is a constant value. This unique characteristic distinguishes hyperbolas from other conic sections such as circles, ellipses, and parabolas.

Step-by-step solution

Definition of Conic Sections

A conic section is a curve obtained as the intersection of the surface of a cone with a plane. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas.

Distinguishing Hyperbolas from Other Conic Sections

In order to identify the property defining a hyperbola, we need to understand how it differs from other conic sections: - In a circle, the distance from the center to any point on the curve is a constant value. - In an ellipse, the sum of distances from the two foci to any point on the curve is constant. - In a parabola, the distance between the focus and any point on the curve is equal to the distance between that point and a line called the directrix.

Properties of Hyperbolas

A hyperbola is defined by the following property: The difference of distances from two fixed points (called foci) to any point on the curve is constant. Mathematically, we can represent this property using the equation: \[|PF_1 - PF_2| = 2a\] Where \(PF_1\) and \(PF_2\) are the distances from the foci to a point P(x, y) on the curve, and 2a is the constant difference in distances. This difference of distances is a defining characteristic of hyperbolas that sets them apart from other conic sections. The shape of a hyperbola consists of two disconnected branches that open either horizontally or vertically, and it helps to understand the unique properties needed to fully describe them, such as asymptotes and the distance between the foci (2c).