Problem 69 - College Algebra

Question

What is an ellipse?

Step-by-Step Solution

Answer

An ellipse is a plane curve with two focal points, where the sum of the distances from any point on the curve to the two focal points is a constant. It has properties such as a center, major and minor axes, and foci. The formula of an ellipse is \((\frac{x-h}{a})^2 + (\frac{y-k}{b})^2 = 1\), where (h, k) is the center of the ellipse, and a and b are the lengths of the major and minor axes.

Step 1: Definition

An ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It is a conic section formed by the intersection of a cone with a plane.

Step 2: Properties

The properties of an ellipse include factors such as its center, major axis, minor axis, foci and the distances between them. The major axis is the longest diameter and the minor axis is the smallest diameter of the ellipse. The foci are two distinct points on the major axis, and each point on the ellipse is equidistant from the foci.

Step 3: Formula

Its standard form of the equation is \((\frac{x-h}{a})^2 + (\frac{y-k}{b})^2 = 1\), where (h, k) is the center of the ellipse, and a and b are the lengths of the major and minor axes, respectively.