Question

a. Show that an equation of the line \(y=m x+b\) in polar coordinates is \(r=\frac{b}{\sin \theta-m \cos \theta}\) b. Use the figure to find an alternative polar equation of a line. \(r \cos \left(\theta_{0}-\theta\right)=r_{0}\) Note that \(Q\left(r_{0}, \theta_{0}\right)\) is a fixed point on the line such that \(O Q\) is perpendicular to the line and \(r_{0} \geq 0\) \(P(r, \theta)\) is an arbitrary point on the line.

Short answer

Question: Convert the given Cartesian equation of a line, \(y = mx + b\), into a polar equation and provide an alternative polar equation using the given figure and line properties. Answer: The polar equation for the given line is \(r = \frac{b}{\sin \theta - m \cos \theta}\). An alternative polar equation based on the given properties and figure is \(r \cos(\theta_0 - \theta) = r_0\).

Step-by-step solution

Express Cartesian Coordinates in Polar Coordinates

To find the polar equation of a line, we first need to express the Cartesian coordinates (x, y) in terms of polar coordinates (r, θ). We use the following transformation formulas: \(x = r \cos \theta\) \(y = r \sin \theta\)

Substitute Polar Coordinates in the Cartesian Equation

Given the Cartesian equation of a line as \(y = mx + b\). We substitute the expressions for x and y from Step 1 into this equation: \(r \sin \theta = m(r \cos \theta) + b\)

Solve for r in terms of θ

From the substituted equation, we isolate r to find the polar equation: \(r(\sin \theta - m \cos \theta) = b\) Solve for r: \(r = \frac{b}{\sin \theta - m \cos \theta}\) This is the polar equation for the given line in part a.

Use the Figure to Find an Alternative Polar Equation for a Line

In part b, we are given the following line properties: - Point Q with (r0, θ0) is a fixed point on the line - Line segment OQ is perpendicular to the line with r0 ≥ 0 - Point P with (r, θ) is an arbitrary point on the line Based on the given properties, we can use the relationship between OQ and PQ in the figure. The angle between OP and PQ is \((\theta_0 - \theta)\). Since PQ is perpendicular to OQ, we have: \(r \cos(\theta_0 - \theta) = r_0\) This is the alternative polar equation for the given line in part b.

Key concepts

Polar Equation of a Line

To express a line in polar coordinates, we start with the Cartesian equation of a line, which is typically written as \(y = mx + b\). The goal is to convert this into a polar equation. In polar coordinates, every point is described by its distance \(r\) from the origin and the angle \(\theta\) from the positive x-axis.

The transformation between Cartesian and polar coordinates involves the identities:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

By substituting these into the line equation, \(y = mx + b\), we get \(r \sin \theta = m(r \cos \theta) + b\).

Simplifying, it becomes \(r(\sin \theta - m \cos \theta) = b\). Solving for \(r\), this gives: \[ r = \frac{b}{\sin \theta - m \cos \theta} \] That's the polar equation of a line in terms of \(r\) and \(\theta\).

Cartesian to Polar Conversion

Converting Cartesian equations to polar form involves using key trigonometric relationships. In polar coordinates, \(x\) and \(y\) are expressed in terms of \(r\) (the radius or distance from the origin) and \(\theta\) (the angle from the positive x-axis).

These conversions are crucial:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

When given a Cartesian line equation, we can use these expressions to transform it into polar coordinates.

This allows us to then manipulate the equation to express \(r\) in terms of \(\theta\), ultimately representing the same line but in polar form.

Conversion strengthens the understanding of different geometrical interpretations and is essential for various applications in physics and engineering.

Trigonometric Identities

When working with polar coordinates, trigonometric identities become essential tools. They allow us to simplify and reframe expressions, crucial when converting from Cartesian to polar forms.

For example, the identity \(\tan \theta = \frac{y}{x}\) links the two coordinate systems by expressing tangent as a ratio of \(y\) over \(x\). This is useful when replacing Cartesian coordinates with polar expressions.

Another key identity is derived from the Pythagorean theorem, where \(r = \sqrt{x^2 + y^2}\). This helps find \(r\) for given \(x\) and \(y\), underpinning the basic polar transformation.

Trigonometric identities provide a bridge between the geometrical aspects of Cartesian and polar systems, enriching our understanding of angles and distances.

Geometry of Lines in Polar Coordinates

In polar coordinates, understanding the geometry of lines requires viewing them differently compared to the Cartesian perspective. Here, lines may be described in terms of a fixed point and an angle.

Consider a line described by the equation \(r \cos(\theta_0 - \theta) = r_0\). Here, \(Q(r_0, \theta_0)\) is a fixed point on the line, and \(OQ\) (a line segment from the origin to \(Q\)) is perpendicular to the line. This means \(r_0\) represents the shortest distance from the origin to the line.

By setting \(Q\) as a pivotal point, any point \((r, \theta)\) on the line maintains this perpendicular orientation.

Such geometric insights allow us to interpret lines not merely as infinite hitches of \(x\) and \(y\), but as objects interplaying in a space defined by direction (angle) and magnitude (distance). This perspective is particularly useful in fields like navigation and robotics, where directionality is crucial.