Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Ellipse} &e=\frac{1}{2}&y=1\end{array}$$

Step-by-Step Solution

Answer

The polar equation for the given ellipse is \( r = \frac{1}{2- \cos(\theta)} \).

Step 1: Understand the Polar Equation of a Conic Section

The general polar equation of a conic section with its focus at the pole is \[r =\frac{ed}{1 \pm e\cos(\theta)}\] where e is the eccentricity, d is the distance from the focus to the directrix, and \(r, \theta\) are the polar coordinates. For an ellipse, you use the 'minus' in the \(\pm\) in the denominator because the eccentricity e is less than 1.

Step 2: Calculate the distance from the focus to the directrix (d)

As per the provided information, the directrix is y = 1. In polar coordinates, this line can be written as \(r\cos(\theta) = 1\). For \(\theta = \pi/2\) (as the conic is symmetric with respect to y-axis), the distance d from the focus to the directrix is calculated as 1, because \(r = d\) at \(\theta = \pi/2\). So, our d is equal to 1.

Step 3: Substitute the values into the Polar Equation of a Conic

Now, we substitute the calculated values - namely, 'e' as 1/2 and 'd' as 1 - into the polar equation of a conic. The equation becomes: \[ r = \frac{\frac{1}{2}*1}{1- \frac{1}{2}\cos(\theta)}= \frac{1}{2- \cos(\theta)}. \]

Key Concepts

Ellipse

An ellipse is a type of conic section that resembles an elongated circle. This shape is formed by the intersection of a plane with a cone, where the angle between the plane and the cone's axis is smaller than the cone's opening angle. This results in a closed curve. Ellipses have two foci, which are integral in their geometric formation.

• A point on the ellipse remains such that the sum of its distances to the two foci is constant.
• In polar coordinates, the focus is at the pole, establishing an elliptical pattern around it.

When describing an ellipse using polar equations, it is important to remember that the conic's eccentricity (e) must always be less than 1. This is a defining trait of ellipses, distinguishing them from other conics like hyperbolas and parabolas.

Eccentricity

Eccentricity is a crucial parameter that determines the shape of a conic section. It is denoted by the symbol \(e\) and it measures how much the conic section deviates from being circular. In the context of ellipses:

• The eccentricity \(e\) is less than 1, which indicates that the shape is an ellipse.
• A smaller value of \(e\) implies a shape closer to a perfect circle.

In the given problem, the eccentricity is \(e = \frac{1}{2}\). This value confirms that the conic is indeed an ellipse, given that it is less than 1. With such an eccentricity, the shape of the ellipse will be fairly circular, but still elliptical in appearance.

Directrix

The directrix is a fixed line used in the description of a conic section. For ellipses, it helps define the polar equation and emphasizes the relationship between the curve and its eccentricity.

• It is positioned perpendicular to the axis of symmetry of the ellipse.
• In polar coordinates, as used in the exercise, it's essential to understand how the directrix relates to the pole.

In this particular exercise, the directrix is given as the line \(y = 1\). Converting this into polar coordinates results in the equation \(r \cos(\theta) = 1\). The distance \(d\) from the conic section's focus to the directrix is a key component in forming the polar equation: \[r = \frac{ed}{1 - e\cos(\theta)}\]. In our example, this distance \(d\) simplifies to 1, aiding in constructing the expression of the ellipse in polar form.