Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ 2 y=-x+4 $$

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{2}{\sin \theta + \frac{1}{2} \cos \theta}\).

Step 1: Rewrite the Cartesian Equation

The given equation is \(2y = -x + 4\). First, solve for \(y\) to get it in the standard linear form: \(y = -\frac{1}{2}x + 2\).

Step 2: Substitute Polar Coordinates

Recall that in polar coordinates, \(x = r\cos \theta\) and \(y = r\sin \theta\). Substitute these into the equation to convert the Cartesian equation to polar form: \(r\sin \theta = -\frac{1}{2} r\cos \theta + 2\).

Step 3: Solve for r

Rearrange the equation to solve for \(r\): \(r\sin \theta + \frac{1}{2} r\cos \theta = 2\). Factor \(r\) out: \(r(\sin \theta + \frac{1}{2} \cos \theta) = 2\). So, \(r = \frac{2}{\sin \theta + \frac{1}{2} \cos \theta}\).

Step 4: Simplify the Polar Equation

The simplification of the polar equation gives \(r = \frac{2}{\sin \theta + \frac{1}{2} \cos \theta}\) as the polar form of the given Cartesian line.

Key Concepts

Cartesian Coordinates

Cartesian coordinates describe the position of a point in a plane using two numbers, typically labeled as \(x\) and \(y\). Imagine an "X" and "Y" axis crossing at a right angle. The spot where they cross is called the origin and is the point \((0, 0)\).- **How it works**: Each point on this grid is described by how far it is along the "X" axis (the horizontal axis) and the "Y" axis (the vertical one). For example, the point \((3, 2)\) means it's 3 units to the right on the "X" axis and 2 units up on the "Y" axis.- **Orientation**: When moving along the "X" axis, right is positive, and left is negative. Along the "Y" axis, up is positive, and down is negative.Using Cartesian coordinates allows you to describe the position of any point in the space, which makes it particularly useful for graphing equations like linear ones, where points need to be plotted in relation to each other to determine the slope and intercepts.

Linear Equations

Linear equations are fundamental expressions that describe a straight line when graphed. One of the simplest forms is the slope-intercept form: \(y = mx + b\).- **Slope (\(m\))**: Represents the steepness of the line, or how much \(y\) changes with \(x\). A positive slope means the line ascends right, while a negative slope descends.- **Y-Intercept (\(b\))**: The point where the line crosses the "Y" axis. This tells you what \(y\) is when \(x\) equals zero.Linear equations are straightforward and easy to graph or convert into other forms. In the original exercise, the Cartesian equation \(2y = -x + 4\) was converted into a linear form \(y = -\frac{1}{2}x + 2\) to identify the slope and y-intercept. Through their simple nature, linear equations can also act as building blocks in more complex operations, such as transformations between coordinate systems.

Polar Equations

Polar equations offer another way of describing points in a plane, using a system based on a distance and an angle rather than vertical and horizontal measurements.- **What it consists of**: In polar coordinates, each point is determined by a radius \(r\) (how far it is from the origin) and an angle \(\theta\) (the direction from the origin). This angle is measured from the positive "X" axis.- **Conversion to Cartesian**: You can easily switch between polar and Cartesian coordinates with the formulas \(x = r\cos \theta\) and \(y = r\sin \theta\). This conversion finds application in specific mathematical problems.In the presented exercise, the Cartesian equation \(y = -\frac{1}{2}x + 2\) was converted to a polar equation \(r = \frac{2}{\sin \theta + \frac{1}{2} \cos \theta}\). This involves substituting the \(x\) and \(y\) values with their polar equivalents, allowing you to study the equation's behavior in terms of rotational symmetry and radial distance, which is especially useful in fields such as physics and engineering.