Problem 82 - Precalculus with Limits

Question

The equation \(r=\dfrac{ep}{1\pm e\ \sin\ \theta}\) is the equation of an ellipse with \(e<1\). What happens to the lengths of both the major axis and the minor axis when the value of \(e\) remains fixed and the value of \(p\) changes? Use an example to explain your reasoning.

Step-by-Step Solution

Answer

When the value of \( p \) changes while \( e \) remains fixed, both the semi-major axis \( a \) and the semi-minor axis \( b \) of the ellipse increase or decrease proportionally to \( p \).

Step 1: Define the Semi-Major and Semi-Minor Axes

The semi-major axis, \( a \), and semi-minor axis, \( b \), of an ellipse are given by the equations: \( a = p / (1 - e^2) \) and \( b = a \cdot \sqrt{1 - e^2} \).

Step 2: Determine the Impact of Changing Semi-Latus Rectum on Semi-major Axis

The semi-major axis increases or decreases linearly in proportion to \( p \). \nAs, \( a=dfrac{p}{1 - e^2} \), it is clear that as \( p \) becomes larger, \( a \) also becomes larger and vice versa, under a fixed \( e \).

Step 3: Determine the Impact of Changing Semi-Latus Rectum on Semi-minor Axis

For the semi-minor axis, if \( a \) increases or decreases, the semi-minor axis \( b \) will also increase or decrease because: \( b= a \cdot \sqrt{1 - e^2} \), implying that \( b \) is directly proportional to \( a \). Thus as \( p \) increases or decreases, \( b \) will follow the same trend under a fixed \( e \).

Key Concepts

Semi-Major Axis Length

In the realm of conic sections, an ellipse shines as a fascinating figure with distinctive lengths known as the semi-major and semi-minor axis. The semi-major axis lends an ellipse its longest diameter, cutting the shape into two symmetrical halves. Think of it as the maximum radius the ellipse has to offer, and this distance plays a critical role in defining the ellipse's shape and size.

Mathematically, the length of the semi-major axis, denoted as \( a \), can be expressed in terms of ellipse's eccentricity \( e \) (a number between 0 and 1 that measures the deviation from being a circle), and the semi-latus rectum \( p \), which is the distance from the center to the curve along a line perpendicular to the major axis. The formula to calculate \( a \) is \( a = \frac{p}{1 - e^2} \). This tells us something quite interesting – as \( p \) changes while \( e \) remains constant, \( a \) directly scales with \( p \). In simpler terms, if you increase \( p \), the semi-major axis stretches out, making the ellipse wider; if you decrease \( p \), it shrinks, squeezing the ellipse narrower.

Semi-Minor Axis Length

Complementing the semi-major axis is its partner in crime, the semi-minor axis, which conveys the shortest diameter of an ellipse. This axis bisects the ellipse into two perfect halves as well, but perpendicularly to the semi-major axis.

The length of the semi-minor axis, symbolized by \( b \), is intrinsically linked to the length of the semi-major axis. Calculating \( b \) is done through the formula \( b = a \times \text{sqrt}{1 - e^2} \). What's noticeable here is the reliance of \( b \) on \( a \); whenever the semi-major axis undergoes a stretch or squeeze, the semi-minor axis follows suit. The proportionality holds true under the steadfast gaze of a fixed eccentricity \( e \). Therefore, if you're altering the value of \( p \) in your elliptical equations while keeping \( e \) steady, you'll witness a corresponding increase or decrease in the length of the semi-minor axis.

Ellipse Equation

The ellipse may appear to be a mysterious curve, but it conforms to a precise mathematical representation. This gives rise to the ellipse equation, a blueprint for plotting this shape on a coordinate plane. The standard form for the equation of an ellipse centered at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) represent the lengths of the semi-major and semi-minor axis respectively.

The curve of an ellipse is well-defined by these parameters, with every point on the ellipse satisfying the equation. If we consider the polar form of the ellipse equation given in the exercise, \( r = \frac{ep}{1 \pm e \sin \theta} \), it amply demonstrates that the ellipse retains its shape due to the interdependence of the radial distance \( r \), the angle \( \theta \), and the constants \( e \) and \( p \). Notably, this form highlights how the semi-latus rectum \( p \) influences the curve – tweak \( p \), and you'll see a proportional adjustment in the overall dimensions of the ellipse, anchored by the firm eccentricity \( e \).

Conic Sections

Ellipses belong to a category of shapes called conic sections, which are created by intersecting a plane with a cone at various angles. In this illustrious family, you'll find circles, ellipses, parabolas, and hyperbolas – each with its unique features depending on the intersection's nature. Ellipses are the result of a slanted cut through a cone that's less steep than the side of the cone but doesn't go through the base; this gives the ellipse its oval shape.

When learning about conic sections, understanding the impact of eccentricity is critical. A circle is a specific type of ellipse with an eccentricity of 0, meaning it's perfectly round. As the eccentricity increases towards 1, the ellipse elongates. In contrast, when you venture beyond an eccentricity of 1, you encounter hyperbolas – open-ended curves that extend infinitely. Parabolas are the sweet spot with an eccentricity exactly equal to 1, opening up symmetrically in one direction. Position, scale, and the measure of \( p \) all depend on where the cutting plane meets the cone, making conic sections an enchanting study of intersecting 3D surfaces and 2D planes.