An ellipse is a geometric shape closely related to a circle but with distinct properties. While a circle consists of all points equidistant from a single center point, an ellipse is defined as the set of all points where the sum of the distances to two fixed points, called the foci, remains constant. This difference means that instead of a single radius, an ellipse involves two key distances that determine its shape.

Ellipses can be oriented either vertically or horizontally. In a vertically stretched ellipse, the foci lie along the y-axis, and the ellipse extends further in the vertical direction. Conversely, in a horizontally stretched ellipse, the foci are positioned along the x-axis, and the ellipse stretches more along the horizontal direction. The two main parameters that describe an ellipse are a and b, representing the distances from the center to the ellipse along the major and minor axes, respectively. For a vertical ellipse, b is the longer distance (vertical stretch), and a is the shorter (horizontal). For a horizontal ellipse, a is the longer distance (horizontal stretch), and b is the shorter (vertical).

The standard equation of an ellipse centered at the origin reflects these distances and is similar in form to the equation of a circle. The equation is given by:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

Here, a and b are the lengths of the semi-major and semi-minor axes, respectively. This equation generalizes the circle's equation, which is a special case where a equals b, representing the radius. For example, if an ellipse is vertically stretched with a = 3 and b = 4, its equation becomes:

$$\frac{x^2}{9} + \frac{y^2}{16} = 1$$

For a horizontally stretched ellipse with a = 4 and b = 3, the equation is:

$$\frac{x^2}{16} + \frac{y^2}{9} = 1$$

When graphing an ellipse centered at the origin, the process involves plotting points a units to the right and left along the x-axis and b units up and down along the y-axis. Connecting these points with a smooth, oval curve forms the ellipse. Understanding the relationship between the ellipse’s foci, axes, and equation is essential for accurately sketching and analyzing ellipses in coordinate geometry.

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