Question

Sketch the given curves and find their points of intersection. $$ r=5, r=\frac{5}{1-2 \cos \theta} $$

Short answer

The curves intersect at the points (0, 5) and (0, -5).

Step-by-step solution

Understand the Polar Equations

The first equation given is a circle in polar coordinates: \( r = 5 \). This represents all points that are 5 units away from the origin.The second equation \( r = \frac{5}{1-2 \cos \theta} \) is a conic section, specifically a limaçon with an inner loop. To sketch these curves accurately, understand the behavior of each in a polar graph.

Graph the Circle

Plot the circle, which is straightforward. A circle with radius 5 centered at the origin can be drawn by marking all points at distance 5 from the center in any direction. In polar coordinates, this spans all angles \( \theta \) while keeping \( r = 5 \).

Analyze and Sketch the Limaçon

The second equation, \( r = \frac{5}{1-2 \cos \theta} \), needs more analysis. The denominator changes with \( \theta \), which affects the value of \( r \) and the resulting shape. When \( \theta = 0 \), \( r = \frac{5}{1-2} = -5 \) (indicating negative direction along the x-axis). Identify other key points by calculating \( r \) for various critical angles (e.g., \( \theta = \pi/2, \pi, 3\pi/2 \)). Sketch the curve using these points, recognizing that the curve loops back through the origin.

Set Equations Equal to Find Intersection Points

To find where these curves intersect, set the two equations equal: \( 5 = \frac{5}{1-2 \cos \theta} \). Simplifying, this leads to:\[ 1 = \frac{1}{1-2 \cos \theta} \]which simplifies further to \( 1 = 1-2 \cos \theta \). Solving for \( \cos \theta \) gives \( \cos \theta = 0 \). Theta values that satisfy this condition are \( \theta = \pi/2 \) and \( \theta = 3\pi/2 \).

Calculate Intersection Radii

For both of these angles, substitute \( \theta \) back into either original polar equation to find the corresponding \( r \). For \( \theta = \pi/2 \) and \( \theta = 3\pi/2 \), substituting into \( r = 5 \) gives \( r = 5 \). Since the circle's \( r \) does not change with angle, the points (\( r, \theta \)) = (5, \pi/2) and (5, 3\pi/2) are intersection points.

Conclusion of the Intersection Points

The coordinates found in polar format correspond to Cartesian coordinates (0, 5) and (0, -5) upon converting \((r, \theta)\) from polar to Cartesian coordinates. These represent the intersection points of the curves on the graph.

Key concepts

Intersection Points

Intersection points occur where two or more curves meet at the same location on a graph. In polar coordinates, this means the same radius and angle define a shared point for multiple curves. To find these points, one typically sets the polar equations equal to each other and solves for \( \theta \).
When considering the equations \( r=5 \) and \( r = \frac{5}{1-2 \cos \theta} \), equating them helps identify the points they have in common.
Solving these equations involves algebraic manipulation, leading to the solution \( \cos \theta = 0 \). This equation holds true for \( \theta = \pi/2 \) and \( \theta = 3\pi/2 \), indicating intersection points at these angles.
Once \( \theta \) is determined, the corresponding \( r \) values are evaluated using either equation to find the exact intersection points in polar form.
These values are vital for sketching polar curves accurately, ensuring precise mapping of their shared points.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone.
These include circles, ellipses, parabolas, and hyperbolas, each with unique characteristics. In polar coordinates, conic sections can take various forms based on the eccentricity and specific parameters of their equations.
The actual conic section involved depends on parameters like the coefficient of \( \cos \theta \). For instance, the provided equation \( r = \frac{5}{1-2 \cos \theta} \) is a type of conic known as a limaçon, specifically one with an inner loop.
Understanding the shape and nature of conic sections can greatly help in identifying how they intersect with other curves or shapes, like a circle given as \( r = 5 \).
Knowing these properties allows for predicting and sketching the shape based on its equation and estimating intersection points with other figures.

Limaçon Curve

A limaçon curve is a fascinating and distinctive type of polar graph. Its defining feature is the presence of loops, influenced by the constant and coefficient values in its equation.
In polar coordinates, the equation \( r = \frac{5}{1-2 \cos \theta} \) describes a limaçon with an inner loop. Limaçons can be categorized based on their loops, such as with inner loops, dimpled, or convex without loops.
The behavior of \( \theta \) in the equation dramatically impacts its shape. For instance, as \( \theta \) changes, the denominator \( 1-2 \cos \theta \) alters the curve's form and orientation.
Exploring these changes results in uniquely looping patterns that can re-cross the origin, characteristic of this type of polar graph.
Sketching a limaçon involves plotting key points and understanding how the loop forms and retracts based on different directional angles.

Polar Graphing

Polar graphing is a method for plotting curves where points are defined by angles \( \theta \) and radius \( r \). Instead of the standard Cartesian \((x, y)\) coordinates, polar graphs use this system to display curves more naturally suited to circular motion.
The circle \( r = 5 \) is straightforward to plot, maintaining a constant radius at any angle \( \theta \). These graphs are symmetrical and provide easy visualization of patterns involving rotational symmetry.
Polar equations, like \( r = \frac{5}{1-2 \cos \theta} \), are graphed by evaluating \( r \) for various \( \theta \) values. As \( \theta \) varies, so does the graph's configuration, resulting in intricate patterns inclusive of loops and intersections.
Understanding polar graph interpretation is crucial for exploring relationships between different polar curves and accurately predicting their behavior on the graph.