Question

Describe what happens to the distance between the directrix and the center of an ellipse if the foci remain fixed and \(e\) approaches 0 .

Short answer

As the eccentricity \(e\) approaches 0, the distance between the directrix and the center of the ellipse increases indefinitely.

Step-by-step solution

Understand the properties of an Ellipse

An ellipse is defined as the set of all points in a plane such that the sum of the distances from two fixed points (the foci) is constant. The formula for eccentricity \(e\) is \(c/a\) where \(c\) is the distance from the center to the focus, and \(a\) is the distance from the center to the vertex. The directrix of an ellipse is a line perpendicular to the major axis such that all points on that line are a fixed proportional distance away from the foci. The formula for directrix \(d\) is \(a/e\).

Analyze the scenario

It's given that the distance between the foci is fixed, indicating that the shape of the ellipse isn't changing. Hence, the value of \(c\) remains constant. However, the value of \(e\) is approaching 0.

Determine the effect on distance between directrix and center

As \(e\) approaches 0, the denominator in the formula for the directrix \(d=a/e\) also approaches 0. Accordingly, the value of \(d\) increases as \(e\) decreases, implying that the distance between the directrix and the center of the ellipse increases as the ellipse tends more towards a circle.

Key concepts

Eccentricity

Eccentricity, symbolized with the lower-case letter ​\( e \)​, is a measure of how much an ellipse deviates from being circular. In basic terms, if an ellipse was a stretched-out circle, eccentricity would tell us how stretched it is. For a perfect circle, the eccentricity is 0 because it's not stretched at all, while for an ellipse, the eccentricity is between 0 and 1. The closer the value of eccentricity is to 1, the more elongated or 'eccentric' the ellipse is.

In the exercise, we see that as the eccentricity approaches 0, the ellipse becomes increasingly similar to a circle. Since the eccentricity is the ratio of the distance from the center to a focus (​\( c \)​) over the distance from the center to a vertex (​\( a \)​), a lower eccentricity means that the ellipse's foci are coming closer to the center, relative to the vertices. This has implications for the appearance and structural properties of the ellipse.

Directrix

A directrix is a concept that might sound complex but is quite intriguing when you delve into it. In the context of an ellipse, a directrix is a straight line that is outside the ellipse itself. Each ellipse has two directrices, which are parallel to the minor axis and equidistant from the center. To reach the formal definition, an ellipse can be defined as the set of all points where the distance to a focus is a constant fraction (less than 1) of the perpendicular distance to the nearest directrix.

The importance of the directrix comes into play when you're computing where points lie on an ellipse, as a complement to the focus-based definition. It's not something visible like the ellipse itself, but an underlying abstract mathematical concept that helps us understand and calculate its shape.

Ellipse Geometry

Ellipse geometry embodies the fascinating characteristics of ellipses in Euclidean space. There are a couple of terms and parts significant for understanding the basics of ellipse geometry. The major and minor axes are the largest and smallest diameters of the ellipse, respectively. The intersections of the ellipse with these axes are called vertices. The longest diameter (the major axis) also gives us the foci — the two special points we've mentioned before.

It's vital to recognize that the sum of the distances from any point on the ellipse to the two foci is constant. This unique property is a defining feature and aids in constructing ellipses using strings and pins, a practical manifestation of their geometric properties. Ellipse geometry plays a role in various fields including astronomy, physics, and engineering, illustrating the interconnectivity of mathematics and the real world.

Foci of an Ellipse

Foci (plural of focus) are among the most intriguing elements in the study of ellipses. These are two fixed points within the ellipse that dictate its shape and size. The defining property of an ellipse is that for any point on its curve, the sum of the distances to the two foci is constant, which in essence, maintains the ellipse's form. This summation of distances is equal to the length of the major axis, which is twice the semi-major axis (​\( 2a \)​).

If you're working to visualize this, think of a string attached to the two pins (foci) and stretched with a pen to draw the ellipse. The string length remains constant, ensuring that all points drawn fulfill the basic property of an ellipse. Fascinatingly, even the orbits of planets are ellipses with the sun positioned at one of the foci, making this concept not merely academic, but also universally significant.