Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{4}{2+2 \quad \sin \theta}\)

Step-by-Step Solution

Answer

Convert polar form to \(\sqrt{x^2+y^2} + y = 2\), a semi-final form. Eliminate the square root via squaring.

Step 1: Identify the Polar Equation

The given polar equation is \(r = \frac{4}{2 + 2 \sin \theta}\). Our goal is to convert this to a rectangular equation in terms of \(x\) and \(y\).

Step 2: Simplify the Polar Equation

Simplify the equation by factoring the denominator: \(r = \frac{4}{2(1 + \sin \theta)} = \frac{2}{1 + \sin \theta}\).

Step 3: Use Polar to Rectangular Conversions

Recall the conversion formulas: \(x = r \cos \theta\), \(y = r \sin \theta\), \(\sin \theta = \frac{y}{r}\). Substitute \(\sin \theta = \frac{y}{r}\) into the equation, giving \(r = \frac{2}{1 + \frac{y}{r}}\).

Step 4: Clear Fractions and Rearrange

Multiply both sides by \(1 + \frac{y}{r}\) to eliminate the fraction: \(r(1 + \frac{y}{r}) = 2\). This simplifies to \(r + y = 2\).

Step 5: Express \(r\) in Terms of \(x\) and \(y\)

From \(r^2 = x^2 + y^2\) and \(r = \sqrt{x^2 + y^2}\), we substitute into the equation: \(\sqrt{x^2 + y^2} + y = 2\).

Step 6: Square Both Sides to Eliminate the Square Root

Square both sides to get rid of the square root: \((\sqrt{x^2 + y^2} + y)^2 = 2^2\). This results in \(x^2 + y^2 + 2y \sqrt{x^2 + y^2} + y^2 = 4\).

Step 7: Solve for a Simplified Rectangular Equation

Set terms involving \(\sqrt{x^2 + y^2}\) aside, resulting in \(x^2 + 2y^2 + 2y \sqrt{x^2 + y^2} = 4\), leading to a specific form dependent on simplification of \(y\).

Step 8: Final Rectangular Equation

The equation generally involves algebraic manipulation to express in a typical conic form, using \(y\) or \(x\) alone, possibly identifying a specific equation form after full expansion and consolidation.

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in a plane using a distance and an angle. Instead of the usual Cartesian system, which uses an x-coordinate and a y-coordinate, polar coordinates use:

• \( r \): The radial distance from the pole (origin) to the point.
• \( \theta \): The angle measured from the positive x-axis.

The point is specified in the format \( (r, \theta) \). This system is particularly handy when dealing with problems that involve circles or rotational symmetry because the position of a point is inherently linked to a central point, making it easier to understand certain conic sections like ellipses, parabolas, and hyperbolas. Polar coordinates also simplify the equations for curves that are not easily expressed in rectangular form. For example, circles centered at the origin or lines that pass through it. To work with these problems, converting to rectangular coordinates can reveal insights not immediately obvious in the polar form.

Rectangular Coordinates

Rectangular coordinates are the typical way to describe the location of a point in a plane using two linear measurements. In this system, each point is defined by an:

• \( x \): The horizontal distance from the origin.
• \( y \): The vertical distance from the origin.

Points are specified as \( (x, y) \). This coordinate system is eminently practical for modeling straight lines and linear equations, making it the standard in algebra and calculus. When polar equations are converted to rectangular equations, we use the relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r = \sqrt{x^2 + y^2} \)
• \( \tan \theta = \frac{y}{x} \)

Rectangular coordinates can make it easier to analyze curves against the horizontal and vertical axis of a graph, which can be especially useful in identifying certain conic sections by their standard equations like the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. The angle and position of the plane result in different types of curves, known as:

• Ellipses
• Parabolas
• Hyperbolas

Each type has distinct properties and equations. For example:

• An ellipse can be formed when the plane cuts through the cone at an angle, but not so steeply to intersect the base. Its general equation in rectangular form is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the distances from the center to the vertices along the x and y axes respectively.
• A parabola occurs when the plane is parallel to the edge of the cone. It has the form \( y = ax^2 + bx + c \) in rectangular coordinates.
• A hyperbola forms when the plane passes through both nappes (the two symmetric parts) of the cone. It can be characterized by \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

These sections are important in physics, astronomy, and engineering. Polar and rectangular forms help in different analyses, with polar coordinates providing simplicity for radial symmetry and rectangular offering straightforward analysis along axes.