Distance Describe what happens to the distancebetween the...

Key Concepts

Eccentricity

Eccentricity is a measure of how much a conic section, such as an ellipse, deviates from being circular. In an ellipse, the eccentricity (a number between 0 and 1) defines the ratio of the distance from any point on the ellipse to one of its foci and the distance from that point to its corresponding directrix. As the eccentricity decreases towards 0, the shape of the ellipse becomes closer to a circle.

Directrix

The directrix of a conic section is a fixed line used in its geometric definition. For an ellipse, the distance from its center to the directrix is directly related to the semi-major axis and the eccentricity. Specifically, this distance increases as the eccentricity decreases, meaning that when the foci are held fixed, the directrix moves farther away from the center as the ellipse becomes more circular.

Fixed Foci and Geometric Relationships

When the foci of an ellipse are fixed, the semi-major axis is determined by the relationship e = c/a, where c is the constant distance from the center to a focus and e is the eccentricity. As the eccentricity approaches 0, the semi-major axis grows larger, which in turn pushes the directrix further from the center. This illustrates how specific geometric relationships within conic sections interconnect the position of the directrix, the ellipse's shape, and its defining parameters.

Transcript

00:03 Okay, so the problem is we want to describe what happens to the distance between the directrix and the center of an ellipse when the foci remain fixed and the eccentricity approaches zero so to be in width let's draw a picture of our ellipse okay, over here we have our ellipse and let's talk a little bit what about our distances are so say this is the center so usually we use this distance over here and we call this b and this will generally be half the diameter in the shorter direction and similarly the half the diameter in the longer direction we're going to call a so now another important measurement is the distance between the center and the focus we'll call this f and this distance is going to be c okay now let's talk a little bit about the directrix which is going to be out here and this is the direct tricks and an useful fact for this problem is that the distance between the center and the directrix can actually be written in terms of a and c.

01:48 And in fact, it's a squared over c.

02:12 So we want to know what happens with this distance here.

02:15 Let's call this x, just for convenience.

02:22 When the foci remain fixed and the eccentricity approaches zero.

02:27 So the foci remaining fixed is essentially saying that c will not change, right? so really, when we're interested to see what happens to a, if c, remains fixed.

02:38 And note that the eccentricity e can be described as c over a.

02:51 Okay, so let's, and what does that mean? well, let's take a look.

02:59 So if c is fixed and e decreases, what happens to a? well, a is going to increase, right? and then we note that if a increases and c is fixed, then what happens to x.

03:35 And that gives us the answer to a problem, right? so, summarizing again, c remains fixed because that determines the distance between the two foci...

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