Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. \(r(3-2 \sin \theta)=6\)

Step-by-Step Solution

Answer

The conic section described is an ellipse; determine focal points based around structure nuances unifying through original form derivation.

Step 1: Recognize the Conic Type

Given the polar equation \( r(3-2 \sin \theta)=6 \), this can be rewritten as \( r = \frac{6}{3 - 2 \sin \theta} \). The structure \( r = \frac{a}{b + c \sin \theta} \) suggests a conic section. Specifically, this is the equation of a conic section in polar form, which can represent parabolas, ellipses, or hyperbolas depending on the coefficients.

Step 2: Convert Polar to Rectangular Form

To identify the type of conic, convert the polar equation to rectangular form. Start with the conversion formulas: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r = \sqrt{x^2 + y^2} \). Substitute these in to find \(r(3-2 \frac{y}{r})=6\), leading to \(3r - 2y = 6\). Simplifying gives \(x^2 + y^2 = \frac{6}{3} + 2y\). Rearranging terms, this equation forms a recognizable conic structure.

Step 3: Simplify and Identify the Conic Type

Simplifying, \(3r = 6 + 2y\) results in \(r = \frac{6 + 2y}{3}\). To see it better, substitute back with \( r^2 = x^2 + y^2\). After simplifying the substitution, recognize this forms a familiar relation outlining a circle centered not at origin, indicating a transformation suggesting ellipse-like properties.

Step 4: Distinguish Between Conic Types

Understand the standard form for ellipses \((\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1)\) and hyperbolas \((\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1)\). The equation does not fit these exactly, hinting that \(3 - 2\sin\theta\) more closely models a circle or ellipse due to symmetry about an axis, but transformed. Hyperbola structures carry subtraction even in orientations.

Step 5: Determine the Foci and Vertices

Review \(r = \frac{6}{3 - 2 \sin \theta}\) previously simplified, yielding symmetry with \(a = 3\), leading to focal distance and semi-major/minor measures. For symmetry evaluation, estimate at traditional angle values, marking visible extrema around or through circles structure per equivalent shape.

Step 6: Graph the Conic and Label Features

Graph the conic section with respect to this understood transformed structure. Label the possible vertices, foci. Confirm calculations correct earlier synthesis steps. Clear points can be derived point testively by substituting theoretical measures through theta cycles and contrast known ellipse stretches.

Key Concepts

Conic Sections

Conic sections are shapes created by slicing a double-napped cone with a plane. Depending on the angle and position of the cut, different shapes can be formed, such as circles, ellipses, parabolas, and hyperbolas. Each of these has unique characteristics and equations that describe them in both Cartesian and polar coordinates.

• *Circles* are formed when the plane slices through the cone parallel to its base, creating a perfectly symmetric shape around its center.
• *Ellipses* appear when the slice is tilted, creating an elongated circle with two foci instead of one central point.
• *Parabolas* are the result when the slice is parallel to the slant of the cone, making a path that curves around a single fixed point, known as the focus.
• *Hyperbolas* happen when the plane cuts through both nappe of the cone, leading to two distinct curves.

Each type of conic section can be described and graphed using a specific equation that highlights its distinct properties. In polar coordinates, equations can help identify which conic section a curve belongs to by examining the coefficients of sine or cosine terms. Identifying the conic involves observing symmetries and analyzing the standard form of its equation.

Graphing Parabolas

Parabolas can be thought of as a symmetrical curve resulting from a cut through a cone parallel to its side. They have some unique properties:

• Parabolas have a single focus—a point where reflections of the curves concentrate.
• The directrix is a line from which distances to a point on the curve are measured, maintaining a consistent relationship with the focus.
• The vertex stands between the focus and the directrix, representing the point of symmetry and maximum curvature.

To graph a parabola from its equation in polar coordinates, convert it to rectangular form using relationships such as \(x = r \cos \theta\) and \(y = r \sin \theta\). As in the original solution, simplification can reveal the characteristics required to plot the parabola correctly. Identifying the axis of symmetry and calculating distances from focus to directrix helps in ensuring accuracy in plotting.

Ellipses and Hyperbolas

When graphing ellipses and hyperbolas, understanding their distinct characteristics is crucial. Both these conic sections involve calculations around axes and foci:

Ellipses

• Ellipses are balanced around two foci. The sum of distances from any point on the ellipse to each focus is constant.
• Equations for ellipses look like \(\left(\frac{x^2}{a^2}\right) + \left(\frac{y^2}{b^2}\right) = 1\), identifying major and minor axes via coefficients.
• Vertices lie on the longer axis, directing the main stretch of the ellipse.

Hyperbolas

• Hyperbolas consist of two branches, each a mirror image opening outwards.
• The equation form \(\left(\frac{x^2}{a^2}\right) - \left(\frac{y^2}{b^2}\right) = 1\) dictates axis separations and orientations.
• Markers like foci are beyond vertices in the directions of the main axis expansion.

For both these shapes, plotting focuses on knowing central points, measuring axis lengths, and assessing relative placements of foci and vertices. This approach ensures clarity in drawing and understanding their geometric contours.