Problem 33 - Algebra and Trigonometry

Question

How are the conics described in terms of a fixed point and a fixed line?

Step-by-Step Solution

Answer

A circle is defined by a set of points equidistant from a fixed center point. An ellipse is defined by the sum of distances to two fixed foci being constant. A parabola is defined by points in a plane equidistant from a fixed focus and a fixed line (directrix). A hyperbola is defined by the absolute difference of distances from two fixed foci being constant and has individual directrices for each branch.

Step 1: Circle Definition

A circle is a set of points in a plane that are all an equal distance from a fixed point. This fixed point is called the center of the circle.

Step 2: Ellipse Definition

An ellipse is a set of points in a plane such that the sum of the distances to two fixed points is constant. These fixed points are called the foci of the ellipse. Unlike a circle, an ellipse does not have a directrix.

Step 3: Parabola Definition

A parabola is a set of points in a plane that are equidistant from a fixed point and a fixed line. The fixed point is called the focus of the parabola and the line is called the directrix. The vertex of the parabola is midway between the focus and directrix.

Step 4: Hyperbola Definition

A hyperbola is a set of points in a plane such that the absolute difference of the distances from two fixed points is constant. These fixed points are the foci of the hyperbola, and the line connecting the foci intersects the hyperbola at two points, which are the vertices of the hyperbola. Each branch of the hyperbola has its own directrix.