Problem 28 - College Algebra

Question

Convert the polar equation of a conic section to a rectangular equation. $$ r(2.5-2.5 \sin \theta)=5 $$

Step-by-Step Solution

Answer

The rectangular equation is \( x^2 + y^2 = 1 \).

Step 1: Expand and Rearrange the Polar Equation

Start with the given polar equation: \[ r(2.5 - 2.5 \sin \theta) = 5 \]. Expand the left-hand side to get: \[ 2.5r - 2.5r \sin \theta = 5 \]. Isolate the term with \( r \sin \theta \) to obtain: \[ 2.5r \sin \theta = 2.5r - 5 \]. Divide through by 2.5 to simplify: \[ r \sin \theta = r - 2 \].

Step 2: Replace Polar Coordinates with Rectangular Coordinates

Recall the conversion formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute \( y \) for \( r \sin \theta \) in the equation derived from Step 1: \[ y = r - 2 \]. Then, express \( r \) in terms of \( x \) and \( y \) using \( r = \sqrt{x^2 + y^2} \), yielding: \[ y = \sqrt{x^2 + y^2} - 2 \].

Step 3: Square Both Sides

To eliminate the square root, square both sides of the equation from Step 2: \[ y^2 = (\sqrt{x^2 + y^2} - 2)^2 \]. This gives: \[ y^2 = x^2 + y^2 - 4\sqrt{x^2 + y^2} + 4 \].

Step 4: Rearrange and Solve for a Rectangular Equation

Move \( x^2 + y^2 \) to the left side to combine like terms: \[ 0 = -4\sqrt{x^2 + y^2} + 4 \]. Solve for \( \sqrt{x^2 + y^2} \): \[ \sqrt{x^2 + y^2} = 1 \]. Square both sides to eliminate the square root: \[ x^2 + y^2 = 1 \].

Step 5: Interpret the Result

The resulting equation \( x^2 + y^2 = 1 \) represents a circle in rectangular coordinates, with its center at the origin (0,0) and a radius of 1.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to describe locations on a plane. Instead of using two perpendicular axes like rectangular coordinates, polar coordinates use a radius and an angle, which can sometimes simplify problems involving symmetry or circular motion. Here are the essentials to grasp:

• The radius \( r \) is the distance from the origin to a point, while the angle \( \theta \) is measured from the positive x-axis.
• A point in the plane is represented as \((r, \theta)\), which specifies its exact location.
• Polar equations often describe curves that are naturally circular or spiral.

By understanding these basic elements, you gain a versatile mathematical tool useful in various applications, from navigation to engineering.

Conic Sections

Conic sections arise when a plane intersects a cone, giving us familiar shapes like circles, ellipses, parabolas, and hyperbolas. These are significant in both pure and applied mathematics.

• A circle, the simplest form of a conic section, is seen here in the transition from polar coordinates to a rectangular form \( x^2 + y^2 = 1 \). This foregrounds a symmetrical relationship around a central point.
• Ellipses and hyperbolas have interesting properties worth exploring but aren't directly applicable to this example.
• Understanding the transformation of a conic section between coordinate systems aids in visualizing their structures and applications.

Conic sections expand our ability to model and solve real-world problems, from physics to graphics programming.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, involve specifying a point's location in terms of two numbers \((x, y)\) along perpendicular axes. This system is intuitive and widely used for various reasons.

• Equations like \( x^2 + y^2 = 1 \) reveal their geometrical nature, which is straightforward to interpret.
• Transforming from polar to rectangular coordinates can simplify equations, making them easier to work with algebraically.
• Such transformations emphasize the grid-like nature of the rectangular coordinate system, facilitating understanding of planar geometry.

Therefore, mastery of rectangular coordinates provides a strong foundation in mathematics that is essential for graphing functions and solving coordinate geometry problems.