Problem 26 - College Algebra

Question

For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r(5+2 \cos \theta)=6 $$

Step-by-Step Solution

Answer

Rectangular form: \(5(x^2 + y^2) + 2x^2 = 6\).

Step 1: Identify Polar Equation Components

Begin with the given polar equation: \[ r(5 + 2 \cos \theta) = 6 \]In this equation, we identify that \(r\) is the distance from the origin to the point, and \(\theta\) is the angle made with the positive x-axis.

Step 2: Solve for r

To make it easier to convert to rectangular coordinates, solve the given equation for \(r\):\[ r = \frac{6}{5 + 2 \cos \theta} \]

Step 3: Apply the Conversion Formulas

Recall the polar to rectangular conversion formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r = \sqrt{x^2 + y^2}\) and \(\cos \theta = \frac{x}{r}\). Use these conversions to transform the polar equation.

Step 4: Substitute \(r = \sqrt{x^2 + y^2}\) and \(\cos \theta = \frac{x}{r}\)

Substitute these into the equation:\[ \sqrt{x^2 + y^2} = \frac{6}{5 + 2 \left(\frac{x}{\sqrt{x^2 + y^2}}\right)} \]

Step 5: Clear the Denominator

Multiply both sides by the denominator to clear it, resulting in:\[ (\sqrt{x^2 + y^2})(5\sqrt{x^2 + y^2} + 2x) = 6 \]

Step 6: Express Equation in Rectangular Coordinates

Simplify the above expression:From step 5, expand and simplify:\[ 5(x^2 + y^2) + 2x\sqrt{x^2 + y^2} = 6 \]

Step 7: Square Both Sides (if necessary)

To eliminate the square root (if steps 5 and 6 don't provide a simple solution), square both sides to isolate terms. Check re-arrangement and consider completing the square or other simplification methods to further reduce.

Step 8: Final Rectangular Form

The equation eventually reduces to rectangular form as:\[ 5(x^2 + y^2) + 2x^2 = 6 \] continuing to simplify will lead to some known form of conic section (ellipse, hyperbola, etc.).

Key Concepts

Polar Coordinates

Polar coordinates are a way to describe the location of points in a plane using two values: a radius and an angle. Unlike rectangular (or Cartesian) coordinates that use the familiar (x, y) format, polar coordinates use (r, θ). Here, 'r' represents the distance from the origin to the point, which is often referred to as the 'radius'. The angle 'θ' (theta) is measured from the positive x-axis to the line segment joining the origin to the point. This is particularly useful in scenarios involving circular or rotational symmetry.

• Radius (r): The distance from the origin.
• Angle (θ): Measured counterclockwise from the positive x-axis.

Polar coordinates provide a simple and intuitive way to express points in scenarios where relationships are naturally circular, such as in trigonometry and certain physical problems.

Rectangular Coordinates

Rectangular coordinates, more commonly known as Cartesian coordinates, represent points in a plane using an ordered pair (x, y). This system is based on two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Every point in this system can be uniquely defined by its horizontal and vertical distances from the origin (0,0).

• x-coordinate: The horizontal distance from the y-axis.
• y-coordinate: The vertical distance from the x-axis.

Rectangular coordinates are highly versatile and can be used in a wide range of mathematical and scientific applications, including geometry, physics, and engineering.

Equation Conversion

Equation conversion between polar and rectangular coordinates involves transforming a given equation into a different coordinate system. This process is crucial in problems where one system is more practical than the other. Common Conversion Formulas:

• To convert from polar to rectangular:
  • Use the relations:
    • x = r \cos(\theta)
    • y = r \sin(\theta)
  • Additionally, \( r = \sqrt{x^2 + y^2} \) and \( \cos(\theta) = \frac{x}{r} \).
• To convert from rectangular to polar:
  • r = \sqrt{x^2 + y^2}
  • \( \theta = \arctan\left(\frac{y}{x}\right) \)

Choosing the suitable coordinate system can vastly simplify the study or solution of the problem at hand.

Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and properties of spaces and figures. In the context of conic sections, geometry helps in understanding the properties and graphs of figures represented by equations in either polar or rectangular coordinates.

• Conic Sections: These include ellipses, parabolas, and hyperbolas, which can be expressed in both polar and rectangular forms.
• Graphs: Understanding the relationship between a point's coordinates and its graphical representation is essential for translating equations between systems.

Comprehending these geometric concepts is vital for solving problems related to the position and shape of objects in a plane.