Question

Find a polar equation for each conic section. Assume one focus is at the origin.

Short answer

Now, create a short answer question based on the information provided: Question: Write the polar equation for a conic section with one focus at the origin, given the eccentricity e and the distance from the origin to the directrix d. Answer: The polar equation for a conic section with one focus at the origin is r = (ed) / (1 + e*cos(θ)).

Step-by-step solution

Ellipses

An ellipse is defined by the sum of the distances from any point on the curve to the two foci is constant. The eccentricity e of an ellipse is between 0 and 1. For this exercise, we are given only one focus. Hence, the polar equation for an ellipse is given by: $r=\frac{ed}{1+e\cos (\theta )}$ where 0 < e < 1, and d is the distance from the origin to the directrix.

Parabolas

A parabola is defined by all points that are equidistant to a focus and a directrix. The eccentricity e of a parabola is exactly 1. The polar equation for a parabola is given by: $r=\frac{d}{1+\cos (\theta )}$ since in this case, e = 1, and d is the distance from the origin to the directrix.

Hyperbolas

A hyperbola is defined as the difference between the distances from any point on the curve to the two foci is constant. The eccentricity e of a hyperbola is greater than 1. The polar equation for a hyperbola is given by: $r=\frac{ed}{1+e\cos (\theta )}$ where e > 1, and d is the distance from the origin to the directrix.

Key concepts

Conic Sections

Conic sections are a group of curves obtained by intersecting a plane with a double cone. They include ellipses, parabolas, and hyperbolas. Each type of conic section has unique properties and equations, and they can be represented in polar coordinates as well as Cartesian coordinates.

To identify a conic section, it's essential to recognize the role of its foci. For ellipses and hyperbolas, there are two foci, while parabolas have one. The position of these foci and how they relate to points on the curve fundamentally define each conic section.

Ellipses are characterized by their closed, oval shape, where the distances from any point on the ellipse to the two foci add up to a constant. Parabolas have a U-shaped curve and are formed by points that are equidistant from a single focus and a directrix. Hyperbolas consist of two separate branches, and the difference in distances from any point on either branch to the two foci is constant.

This geometric understanding of conic sections lays the foundation for forming their equations in different coordinate systems.

Eccentricity

Eccentricity is a number that describes how much a conic section deviates from being circular. It captures the shape and nature of the conic section.

For ellipses, the eccentricity ($e$) is between 0 and 1. As the eccentricity approaches 0, the ellipse becomes more circular. As it gets closer to 1, the ellipse elongates. When $e=1$, the conic is a parabola, which means the curve is open and neither closed like a circle nor elongated like an ellipse.

Hyperbolas have an eccentricity greater than 1. The further the eccentricity exceeds 1, the wider the branches of the hyperbola become. The concept of eccentricity helps in identifying and classifying these curves in polar coordinates by their degree of openness or closeness.

Polar Equations

Polar equations express the positions of points using the polar coordinate system, which relies on a radius and angle from a fixed point, the pole (or origin). For conic sections, the pole typically acts as a focus, and the directrix is a line based on which the curve's exact position is determined.

To describe an ellipse using polar coordinates, the polar equation is $r=\frac{ed}{1+e\cos (\theta )}$, where $0<e<1$. Here, $r$ is the radius, $d$ the distance from the origin to the directrix, and $\theta$ the angle.

The polar equation for a parabola is simpler: $r=\frac{d}{1+\cos (\theta )}$, with the eccentricity $e$ equal to 1, making it a unique case among the conic sections.

For hyperbolas, the polar equation takes a form similar to that of ellipses: $r=\frac{ed}{1+e\cos (\theta )}$, but with $e>1$, indicating a larger eccentricity and more open curve.

Polar equations simplify the interpretation of conic sections, providing a mathematical description based on radius and angle, which are inherent in the geometric nature of these curves.