Problem 4 - Algebra and Trigonometry

Question

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Hyperbola, eccentricity \(\frac{4}{3},\) directrix \(x=-3\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{4}{1 - \frac{4}{3} \cos \theta} \).

Step 1: Write the general polar equation

The general polar equation for a conic with a focus at the origin is given by \( r = \frac{ed}{1 - e \cos \theta} \) where \( e \) is the eccentricity and \( d \) is the distance to the directrix. In our case, \( e = \frac{4}{3} \) and the directrix is \( x = -3 \).

Step 2: Calculate the directrix distance (d)

Since the directrix is \( x = -3 \), the distance \( d \) from the origin to the directrix is \( 3 \). We take the absolute value of the directrix distance since the directrix is a line parallel to the y-axis, not considering any specific direction.

Step 3: Plug values into the polar equation

Substitute the values of \( e = \frac{4}{3} \) and \( d = 3 \) into the polar equation. Thus, we have: \( r = \frac{\frac{4}{3} \times 3}{1 - \frac{4}{3} \cos \theta} \).

Step 4: Simplify the polar equation

Simplify the expression: \( r = \frac{4}{1 - \frac{4}{3} \cos \theta} = \frac{4}{1 - \frac{4}{3} \cos \theta} \). Simplified, the polar equation becomes \( r = \frac{4}{1 - \frac{4}{3} \cos \theta} \).

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double cone. Depending on the angle and position of the intersection, we get different types of conic sections, namely circle, ellipse, parabola, and hyperbola.

• A **circle** forms when the plane cuts the cone parallel to its base.
• An **ellipse** is obtained when the cut is at an angle, but not parallel to the base.
• A **parabola** arises when the plane is parallel to a slant edge of the cone.
• Finally, a **hyperbola** is formed when the plane intersects both nappes (sides) of the double cone.

Each of these curves has distinct properties that can be described in a coordinate system. In the context of polar coordinates, these curves can be defined by specific equations. Conics defined in polar form have a focus at the origin, expressed by the general formula \( r = \frac{ed}{1 - e \cos \theta} \). Each curve's distinct shape is controlled by parameters like eccentricity \( e \) and distance to the directrix \( d \), which we will explore next.

Eccentricity

Eccentricity is a key parameter in defining the shape of a conic section. It is denoted by \( e \) and determines how much the conic section deviates from being circular.

• When \( e = 0 \), the conic is a circle.
• For \( 0 < e < 1 \), the conic is an ellipse.
• If \( e = 1 \), it's a parabola.
• For \( e > 1 \), we get a hyperbola.

In our exercise, the given eccentricity is \( \frac{4}{3} \), indicating that the conic is a hyperbola since \( e > 1 \). The higher the eccentricity, the "flatter" the conic section looks. For hyperbolas, this implies the two branches are more spread out. Eccentricity is crucial as it influences the equation's behavior in polar coordinates, directly affecting the polar curve's shape.

Directrix

The directrix of a conic section is a fixed line used in the geometric definition of the conic. In polar coordinates, the distance from the pole to this line, denoted by \( d \), is integral in forming the conic's equation.
The role of the directrix is to maintain the ratio of distances: a point on the conic is such that the ratio of its distance to the focus is constant in relation to the directrix. This ratio is the eccentricity \( e \).
In our problem, the directrix is given as \( x = -3 \). This means the directrix is a vertical line located three units to the left of the pole (origin). As this line is used to calculate \( d = 3 \) in the positive sense, it tool helps to build the polar equation for the hyperbola. This combination of factors contributes to the properties and unique traits of the polar graph of the hyperbola.