Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r(2-\cos \theta)=1\)

Step-by-Step Solution

Answer

The rectangular equation is \(3x^2 + 4y^2 - 2x - 1 = 0\).

Step 1: Express polar equation with known identities

Start with the polar equation \( r(2 - \cos \theta) = 1 \). We know that \( r = \sqrt{x^2 + y^2} \), \( x = r \cos \theta \) and \( y = r \sin \theta \). Rewrite \( \cos \theta \) in terms of \( x \) and \( r \): \( \cos \theta = \frac{x}{r} \).

Step 2: Substitute cosine and simplify

Substitute \( \cos \theta \) with \( \frac{x}{r} \) in the equation: \[ r(2 - \frac{x}{r}) = 1 \] Simplify to obtain: \[ 2r - x = 1 \]

Step 3: Express in terms of \( r \) and solve for \( x \)

From \( 2r - x = 1 \), solve for \( r \): \[ 2r = x + 1 \] Thus, \[ r = \frac{x + 1}{2} \].

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

Substitute \( r \) back with \( \sqrt{x^2 + y^2} = \frac{x + 1}{2} \): \[ \sqrt{x^2 + y^2} = \frac{x + 1}{2} \]

Step 5: Eliminate square root by squaring both sides

Square both sides to eliminate the square root: \[ x^2 + y^2 = \left( \frac{x + 1}{2} \right)^2 \] Therefore, \[ x^2 + y^2 = \frac{(x + 1)^2}{4} \].

Step 6: Simplify and arrange into standard form

Multiply through by 4 to clear the fraction: \[ 4(x^2 + y^2) = (x + 1)^2 \] Expand \((x + 1)^2\): \[ (x + 1)^2 = x^2 + 2x + 1 \] Thus, the equation becomes: \[ 4x^2 + 4y^2 = x^2 + 2x + 1 \]

Step 7: Rearrange to standard conic form

Subtract \( x^2 + 2x + 1 \) from both sides: \[ 3x^2 + 4y^2 - 2x - 1 = 0 \] Reorganize the terms if needed, to obtain the final rectangular form.

Key Concepts

Polar Coordinates

Polar coordinates offer a method of representing points in a two-dimensional plane using a radius and an angle, rather than the traditional Cartesian coordinate system of x and y axes. In polar coordinates, a point is defined by the distance from a fixed central point, known as the pole (similar to the origin in the rectangular coordinate system), and the angle from a reference direction, often the positive x-axis. This form is especially useful in problems involving curves and angles, as well as in fields like physics and engineering where circular motion is analyzed.

• Radius (r): Represents the distance from the pole.
• Angle (θ): The measure of rotation from the reference direction.

This system transforms how we perceive graphing functions that are primarily circular or spiral in nature. The ease of describing complex shapes with simple equations in polar form is one of its primary advantages.

Rectangular Coordinates

When dealing with equations of lines, circles, or any graphable function, the Cartesian coordinate system is most familiar. Here, any point in the plane is described with a simple ordered pair of numbers \(x, y\). This coordinate system is crucial because it allows one to easily perform operations like addition or subtraction of coordinates and to apply calculus concepts.
Yet, situations arise where rectangular coordinates might not be the most efficient or intuitive reference system, especially in circular or periodic phenomena. This is where polar coordinates offer a simpler way to describe complex figures, and vice versa, where transitioning from polar to rectangular can facilitate solving equations in real-world applications.

Conic Sections

Conic sections represent curves obtained by slicing a cone at different angles, producing circles, ellipses, parabolas, and hyperbolas. These shapes are vital in various math and science fields because they naturally describe a wide range of phenomena. In converting between polar and rectangular equations, you might encounter these conic sections. The process involves expressing polar equations in terms of x and y.

• Circles: Formed when the cone is sliced perpendicular to the base.
• Ellipses and Hyperbolas: Created by slanting the slice more or less steeply.
• Parabolas: Achieved when the plane is parallel to the slope of the cone.

Understanding the relationship between the form of the equation and its graphical representation is crucial in recognizing these conic sections within different coordinate systems.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the included variables. They are instrumental when converting between polar and rectangular equations.
In the provided problem, the \( \cos \theta = \frac{x}{r} \) identity plays a key role. By understanding these identities, students can bridge the gap between the two coordinate systems.

• Sine and Cosine: Fundamental in defining the relationship between angles and the sides of right-angled triangles.
• Basic Identities: Like \( \sin^2 \theta + \cos^2 \theta = 1 \), essential in manipulating and converting equations.

Trigonometric identities provide the tools needed to seamlessly shift expressions and uncover the equations' inherent geometric properties.