Problem 26 - College Algebra

Question

Convert the polar equation of a conic section to a rectangular equation. $$ r(5+2 \cos \theta)=6 $$

Step-by-Step Solution

Answer

The rectangular form is \( 21x^2 + 25y^2 + 24x = 36 \).

Step 1: Identify the Polar Equation

The given polar equation of the conic is \( r(5 + 2\cos\theta) = 6 \). You need to convert this equation into rectangular form.

Step 2: Expand the Polar Equation

Start by expanding the equation: \( r \times 5 + r \times 2\cos\theta = 6 \). This simplifies to \( 5r + 2r\cos\theta = 6 \).

Step 3: Substitute Polar to Rectangular Relations

Remember that \( r^2 = x^2 + y^2 \) and \( r\cos\theta = x \). Substitute these into the expanded equation: \( 5r + 2x = 6 \).

Step 4: Express \( r \) in terms of \( x \) and \( y \)

From the relation \( r = \sqrt{x^2 + y^2} \), substitute \( r \) in the equation: \( 5\sqrt{x^2 + y^2} + 2x = 6 \).

Step 5: Isolate \( r \)

Rearrange the equation to isolate the \( \sqrt{x^2 + y^2} \): \( 5\sqrt{x^2 + y^2} = 6 - 2x \).

Step 6: Square Both Sides

Square both sides to eliminate the square root: \( 25(x^2 + y^2) = (6 - 2x)^2 \).

Step 7: Expand and Simplify

Expand the right-hand side: \( 25(x^2 + y^2) = 36 - 24x + 4x^2 \). Rearrange the equation: \( 21x^2 + 25y^2 + 24x = 36 \).

Step 8: Final Rectangular Equation

The equation \( 21x^2 + 25y^2 + 24x = 36 \) is the required rectangular form of the given polar equation.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. They are fundamental in geometry and can be classified into four types: circles, ellipses, parabolas, and hyperbolas. Each of these shapes is defined by a specific set of equations, which can be expressed in both polar and rectangular coordinates.
Conic sections play an important role in physics, astronomy, and engineering as they describe planetary orbits, the design of telescopes, and more.

• Circles: Formed when the plane intersects the cone parallel to its base.
• Ellipses: Occur when the plane cuts through the cone at an angle, not perpendicular.
• Parabolas: Result from the plane being parallel to the edge of the cone.
• Hyperbolas: Appear when the plane intersects both nappes of the cone.

Recognizing these forms is crucial as each conic can represent different physical phenomena. Understanding how to convert between coordinates helps to better utilize conic sections in various applications.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use a grid system to define the position of a point in a plane. The coordinates are expressed as \( (x, y) \), where \( x \) represents the horizontal position and \( y \) represents the vertical position.
This system is universal and widely used in graphing, providing a straightforward way to plot equations and visualize geometrical shapes.
Important features of rectangular coordinates include:

• The origin \( (0, 0) \) is the point where the x-axis and y-axis intersect.
• The x-axis is the horizontal line, and the y-axis is the vertical line.
• Positive and negative values of \( x \) and \( y \) indicate the position of a point relative to the origin.

In the task of converting polar equations to rectangular form, the knowledge of rectangular coordinates is essential as it allows us to use algebraic techniques to express complex curves in a simpler and more familiar form.

Polar Coordinates

Polar coordinates represent a point in a plane using a distance and an angle. A point is defined by \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis.
This system is especially useful in situations with circular or radial symmetry, making it ideal for problems involving rotation and spirals.
Key elements of polar coordinates are:

• Radius \(r\): Measures how far the point is from the origin. It is always non-negative.
• Angle \(\theta\): Measured in radians or degrees, it indicates the direction from the origin to the point.
• Conversion to rectangular coordinates is facilitated by the relations \(r\cos\theta = x\) and \(r\sin\theta = y\).

Using polar coordinates, we often describe curves more simply. When converting to rectangular coordinates, these properties help translate polar equations into a form that is more aligned with Cartesian graphing methods.