Question

Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{6}{3+2 \sin \theta}$$

Short answer

Answer: The given polar equation represents a hyperbola. The vertices are located at $$(\frac{9}{4}, 0)$$ and $$(\frac{9}{4}, \pi).$$ The foci are at $$(\frac{27}{4}, 0)$$ and $$(\frac{27}{4}, \pi).$$ The directrices are the vertical lines $$x = \pm \frac{9}{4},$$ and the asymptotes are given by the equations $$y = \pm \frac{9}{4}.$$

Step-by-step solution

Determine the type of conic section

First, we will rewrite the given equation in the form: $$r=\frac{ep}{1+e\sin \theta},$$ where $$e$$ is the eccentricity and $$p$$ is the semi-latus rectum. Comparing with the given equation, $$r=\frac{6}{3+2 \sin \theta},$$ we can find that $$e\cdot p=6$$ and $$(1+e)\cdot p=6+4\cdot p.$$ Solving for $$e$$ and $$p$$, we get $$e=\frac{4}{3}$$ and $$p=\frac{9}{2}.$$ Since $$e>1$$, the conic section is a hyperbola.

Identify vertices, foci, directrices, and asymptotes

For a hyperbola, the semi-latus rectum $$p$$ is related to the distance between the foci $$2f$$ and the distance between the vertices $$2a$$ through $$p=a\cdot e$$ and $$f=a\cdot e.$$ From the calculated values of $$e$$ and $$p$$, we have $$a=\frac{9}{4}$$ and $$f=\frac{27}{4}.$$ The vertices are at polar angles $$0$$ and $$\pi$$, with the polar radius equal to $$a=\frac{9}{4}.$$ Thus, the vertices are $$(\frac{9}{4}, 0)$$ and $$(\frac{9}{4}, \pi).$$ The foci are also at polar angles $$0$$ and $$\pi$$, with the polar radius equal to $$f=\frac{27}{4}.$$ Thus, the foci are $$(\frac{27}{4}, 0)$$ and $$(\frac{27}{4}, \pi).$$ The directrices are vertical lines parallel to the initial ray at a distance $$\frac{p}{e} = \frac{9}{4}$$ on both sides of the pole. Thus, the directrices are $$x = \pm \frac{9}{4}.$$ The asymptotes are lines that the branches of the hyperbola approach but never intersect, located at polar angles $$\frac{\pi}{2}$$ and $$\frac{3\pi}{2}.$$ The equations of the asymptotes are $$r\sin \theta = \pm \frac{9}{4}$$ or $$y=\frac{r}{\sin \theta} = \pm \frac{9}{4}.$$

Plot the conic section

To graph the conic section, follow these steps: 1. Draw the directrices at $$x = \pm \frac{9}{4}.$$ 2. Plot the vertices at $$(\frac{9}{4}, 0)$$ and $$(\frac{9}{4}, \pi).$$ 3. Plot the foci at $$(\frac{27}{4}, 0)$$ and $$(\frac{27}{4}, \pi).$$ 4. Sketch the hyperbola branches symmetrically with respect to the pole and approaching the asymptotes. Then, use a graphing utility to verify the plot. The polar equation $$r=\frac{6}{3+2 \sin \theta}$$ should produce a hyperbola with the calculated vertices, foci, directrices, and asymptotes.

Key concepts

Polar Equations

Polar equations are representations of curves on a two-dimensional plane using the polar coordinate system. This system uses a fixed point called the pole (analogous to the origin in Cartesian coordinates) and a ray from the pole in the direction of the positive x-axis known as the initial ray or polar axis. The position of a point is determined by a distance from the pole, denoted as 'r', and the angle 'θ' formed by the ray to that point and the initial ray.

Polar equations like \( r = \frac{6}{3 + 2 \sin \theta} \) define the relationship between 'r' and 'θ', allowing us to plot the curve point by point. To graph such equations, one typically calculates several pairs of (r, θ) values to draw the shape of the curve on the polar grid. Understanding polar equations leads to easily graphing complex shapes, such as spirals, circles, and in this particular case, conic sections like a hyperbola.

Hyperbola Characteristics

A hyperbola is one of the four basic types of conic sections, defined by its eccentricity (e), which is always greater than 1. Eccentricity measures how much a conic section deviates from being circular, and for a hyperbola, it indicates the shape's 'spread'.

Characteristics distinguishing a hyperbola include:

• Two separate branches that extend to infinity without ever intersecting.
• Two fixed points called foci, around which the hyperbola is shaped.
• Two lines called asymptotes that the branches approach indefinitely but never cross.
• A pair of vertices, which are the closest points to the center along the principal axis of the hyperbola.

Each pair of corresponding points on the two branches are equidistant to the foci, leading to the reflection property that characterizes hyperbolas. The distances from any point on a hyperbola to the foci have a constant difference, which is the source of their distinctive shape.

Vertices and Foci

For conic sections, vertices and foci are critical points that define their shape and orientation. In hyperbolas, the vertices are points where the hyperbola intersects its principal axis, and they are the closest points on the curve to the center. The distance from the center to a vertex is denoted as 'a'.

The foci (plural of focus) of a hyperbola are two fixed points located along the principal axis, both lying outside the curve. They are used to construct the hyperbola since the difference in the distance from any point on the hyperbola to each focus is constant. This constant difference is '2a', where 'a' is the distance from the center to either vertex. For hyperbolas, the distance from center to focus is denoted as 'f' and calculated using the relationship \( f=a\cdot e \), where 'e' is the eccentricity.

Directrices and Asymptotes

In the study of conic sections, directrices and asymptotes are essential concepts to understand their geometries and guide their plotting.

The directrix of a conic section is a line to which the distance of any point on the conic is related by the eccentricity. For hyperbolas, there are two directrices, which serve as reference lines perpendicular to the principal axis. They are located symmetrically on either side of the center and the distance to the directrix from the pole, coupled with the eccentricity, determines the shape of the hyperbola.

Asymptotes, on the other hand, are imaginary lines that a hyperbola tends towards but never actually meets. They intersect at the hyperbola's center and form an 'X' shape that provides a skeleton for drawing the two separate branches of the hyperbola. Knowing the slope and position of the asymptotes is crucial for graphing a hyperbola accurately, as they define the ultimate direction of the curves' branches as they extend towards infinity.