Question

For the following exercises, find a polar equation of the conic with focus at the origin and eccentricity and directrix as given. $$\text{ Directrix: }x=-4;e=5$$

Short answer

The polar equation is $r=\frac{20}{1+5\cos \theta }$.

Step-by-step solution

Identify the Type of Conic

Since the eccentricity $e=5$ is greater than 1, the conic is a hyperbola. In polar coordinates, hyperbolas have the form $r=\frac{ed}{1-e\cos \theta }$ or $r=\frac{ed}{1+e\cos \theta }$ depending on the position of the directrix.

Determine the Position of the Directrix

The directrix is given as $x=-4$, which is to the left of the pole (the origin) along the horizontal axis. For a directrix along the line $x=c$, if $c<0$, the equation uses $+e\cos \theta$. Hence, we will use $r=\frac{ed}{1+e\cos \theta }$.

Calculate Distance from Focus to Directrix

Calculate the distance $d$ from the pole (the origin) to the directrix, which is $|-4|=4$.

Plug Values into the Equation

Now that we have identified that $e=5$ and $d=4$, we can substitute these into the formula. The polar equation becomes: $$r=\frac{5\times 4}{1+5\cos \theta }$$ This simplifies to: $$r=\frac{20}{1+5\cos \theta }$$

Key concepts

Conic Sections

Conic sections are curves that are generated by the intersection of a plane and a cone. These include circles, ellipses, parabolas, and hyperbolas, each defined by their unique properties.
Every conic section can be represented in different coordinate systems, such as Cartesian and Polar.
In the context of polar coordinates, conic sections have forms that use a combination of the angle $\theta$ and the radius $r$. Understanding these shapes:

• **Circle** - A circle in polar form has the equation $r=e$ where $e=0$.
• **Ellipse** - An ellipse has an eccentricity $0<e<1$.
• **Parabola** - A parabola has an eccentricity $e=1$.
• **Hyperbola** - A hyperbola has an eccentricity $e>1$, like in the given problem.

Recognizing the type of conic section is important as it determines the structure and focus of the equation we use.

Eccentricity

Eccentricity is a measure that distinguishes different conic sections. It indicates how much a conic section deviates from being circular.
The value of eccentricity $e$ helps to identify the type of the conic:

• For a circle, $e=0$. This indicates it's a perfectly round shape.
• For an ellipse, $e$ is between 0 and 1, which means it's more stretched out.
• For a parabola, $e=1$. Parabolas are a special case of conics.
• For a hyperbola, like in our problem, $e>1$, showing a significant deviation from a circle and represented in the equation as $r=\frac{ed}{1\pm e\cos \theta }$.

In the given exercise, since $e=5$, the conic is confirmed as a hyperbola, and it has significant stretching.

Hyperbola

A hyperbola is a type of conic section characterized by its open curve with two different branches. These curves appear when the eccentricity $e$ is greater than 1.
In the polar coordinate system, a hyperbola can have equations of the form $r=\frac{ed}{1-e\cos \theta }$ or $r=\frac{ed}{1+e\cos \theta }$. The choice between these forms depends on the orientation and position of the directrix. The hyperbola's unique features include:

• It has two symmetric branches that open away from each other.
• The focuses (plural of focus) of a hyperbola are located outside the vertices of the branches.
• This shape can be found in various natural and scientific phenomena, such as orbits and radio signals.

The exercise helps us understand how to use these properties in the context of polar coordinates.

Directrix

In conic sections, the directrix is a fixed line used along with the focus to define and shape the curve. The relationship between the focus and directrix is intrinsic as it helps determine the conic's eccentricity and orientation. For the hyperbola in this exercise, a directrix such as $x=-4$ is given. This position impacts how the polar equation is formulated. When the directrix is horizontal, either left or right of the origin (the pole), the positive or negative sign in the equation may change:

• If the directrix is $x=c$ where $c<0$, like $x=-4$, the equation used is $r=\frac{ed}{1+e\cos \theta }$.
• Alternatively, if the line were $x=c$ with $c>0$, then the minus sign applies in the equation $r=\frac{ed}{1-e\cos \theta }$.

Understanding the role of the directrix helps in accurately defining the polar equation of a conic, as seen in the current exercise where this principle was applied.