Question

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r(2+\sin \theta)=4\)

Short answer

The eccentricity \(e\) is 1, indicating a parabola. The distance from the pole to the directrix \(d\) is 4. The graph of the equation is a downward-opening parabola.

Step-by-step solution

Rearrange the Polar Equation

Reorganize the equation \(r(2+\sin \theta) = 4\) by dividing through by the \(2+\sin \theta\) term to isolate \(r\), getting \(r = \frac{4}{2 + \sin \theta}\). To identify the shape, compare this with the standard form for conics in polar coordinates, \(r = \frac{ed}{1 \pm e \cos \theta}\) or \(r = \frac{ed}{1 \pm e \sin \theta}\).

Determine the Eccentricity and Directrix

By comparing \(r = \frac{4}{2 + \sin \theta}\) with \(r = \frac{ed}{1 \pm e \sin \theta}\), it can be seen that \(e = 1\) and \(ed = 4\). Using the eccentrity \(e = 1\) and substitute it into \(ed = 4\) to find the value of \(d\), gives us \(d = 4\). Hence, the eccentricity \(e = 1\), and the distance from the pole to the directrix \(d = 4\).

Identify the Graph Type and Sketch the Graph

The eccentricity \(e = 1\) indicates a parabola. Also, the conic opens downwards since the equation is in terms of \(\sin \theta\). You can further clarify this by plotting the sketch graph with a graphing utility. This plot should indicate that the graph of the given equation is indeed a downward-opening parabola.

Key concepts

Eccentricity of a Conic

Understanding the eccentricity of a conic is crucial to identifying and distinguishing between different conic sections—circles, ellipses, parabolas, and hyperbolas. Eccentricity is a number, often denoted as 'e', which uniquely characterizes the shape of a conic section.

Eccentricity values interpret the conic's shape as follows:

• If the eccentricity is equal to zero, \(e = 0\), the conic is a circle.
• If the eccentricity is between zero and one, \(0 < e < 1\), the conic is an ellipse.
• If the eccentricity is exactly one, \(e = 1\), the conic is a parabola.
• If the eccentricity is greater than one, \(e > 1\), the conic is a hyperbola.

In our exercise, the conclusion that the eccentricity \(e = 1\) immediately tells us that the graph represents a parabola. This distinct value is the defining characteristic of parabolic shapes and informs us that the distance from any point on the conic to the focus is equal to the distance from that point to the directrix.

Distance from the Pole to the Directrix

In polar coordinates, the directrix is a fixed reference line used to define a conic, and the distance from the pole (origin) to the directrix is denoted as 'd'. This distance impacts the overall shape and size of the conic section.

For parabolas, where eccentricity \(e = 1\), the formula simplifies, and the directrix is related to the denominator of the polar equation. In the provided exercise, the polar equation \(r(2 + \(\sin \theta\)) = 4\) relates to the standard parabolic form \(r = \frac{ed}{1 + \sin \theta}\), revealing that the distance to the directrix \(d = 4\). This means that every point on the parabola is equidistant from the focus and a line 4 units away from the pole, and it is this concept of equidistance that is fundamental to the definition of a parabola in polar form.

Sketching Conic Sections

Sketching conic sections in polar coordinates requires understanding the underlying equations and how they correspond to the shape of the graph. After determining the type of conic based on its eccentricity and distance to the directrix, one can start to sketch the graph.

The equation \(r(2 + \sin \theta) = 4\) simplifies to a form which tells us the conic will be a parabola with its vertex at the pole and opening downward, as indicated by the \(\sin \theta\) term.

To start sketching the parabola from our exercise, plot the focus at the pole (origin), and draw the directrix line parallel to the horizontal axis at a distance 'd' below it. Next, ensure that each point on the curve maintains an equidistant from the focus and the directrix. This approach guarantees that the resulting sketch reflects the unique properties of a downward-opening parabola in polar coordinates, providing a visual aid that complements analytical understanding.